The mystery of Riemann's curvature

Olivier Darrigol $^{a, b, *}$

^{a}

SPHere, CNRS, France

b OHST, UC-Berkeley, USA

Available online 16 May 2014

Abstract

Bernhard Riemann's habilitation lecture of 1854 on the foundations of geometry contains a stunningly precise concept of curvature without any supporting calculations. Another memoir of 1861 contains formulas in which we may recognize our Riemann tensor, though in a different context and without much geometrical interpretation. The first text is mysterious by the lack of formulas, the second by the excess of formulas. The purpose of this essay is to investigate this double mystery and the stimulating effect it had on some of Riemann's early readers, from Richard Dedekind to Tullio Levi-Civita. Use is made of some heretofore unexploited manuscript sheets by Riemann.

Résumé

La leçon d'habilitation que Bernhard Riemann prononça en 1854 sur les fondements de la géométrie contient un concept de courbure étonnamment précis, sans aucun calcul pour le soutenir. Un autre mémoire de 1861 contient des formules parmi lesquelles le lecteur moderne reconnaît le tenseur de Riemann, cela dans un contexte différent et presque sans interprétation géométrique. Le premier texte est mystérieux par le manque de formules, le second par l'excès de formules. Le but de cet essai est d'examiner ce double mystère et l'effet stimulant qu'il eut sur quelques lecteurs de Riemann, de Richard Dedekind à Tullio Levi-Civita. Quelques notes manuscrites inédites de Riemann sont utilisées à cet effet.

MSC: 01A55; 53-03

Keywords: Riemann; Riemannian geometry; Curvature; Differential invariants; Gauss; Levi-Civita

I still have vivid memories of the extraordinary impression that Riemann's trains of thought made on young mathematicians [when his habilitation lecture was published]. Much of it seemed obscure and hard to understand and yet of unfathomable depth. Today's mathematicians, who have integrated all these things in their way of thinking, still admire the clarity and fertility of the analysis.

^{1}

[Felix Klein, 1926-1927, 2: 16]

0. Introduction

When, in his habilitation lecture of 1854, Bernhard Riemann characterized curvature in a metric manifold of any dimension, he left much for mathematicians and historians to wonder about. The concept there occurred with no calculations whatsoever and yet with stunning precision. Riemann was addressing a mostly non-mathematical audience, the Göttingen faculty of philosophy, who would not have followed the mathematical technicalities. Old Carl Friedrich Gauss probably was the only listener to understand the extraordinary profundity of the lecture. Riemann never published his reflections nor any relevant calculation, because poor health prevented him to do so (according to Dedekind) or maybe because his interest in geometry was only occasional.

^{2}

In 1861, Riemann sent to the French Academy of sciences a memoir usually called the Commentatio, in which the modern reader easily recognizes the precise expression of what is now called the Riemann tensor as well as an argument for the covariance of this tensor. It is tempting to relate the relevant section of this memoir to the geometric problem of finding a criterion of flatness for a manifold, in the wake of the habilitation lecture. Yet the context of the Commentatio was different: it answered a prize question on the propagation of heat in solid bodies. Its derivations were purely analytic, and its only contact with geometry was in Riemann's remark that some analytical expression could be illustrated as the curvature components of a metric manifold. The prize was not attributed, and Riemann's submission rested in the archive of the French Academy.

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Riemann's habilitation lecture and the Commentatio were published posthumously, the former by Richard Dedekind in 1867, and the latter by Heinrich Weber in 1876 in Riemann's Werke. Riemann was no longer there to answer the following questions: How did he obtain the results enunciated in the habilitation lecture? Surely he must have done some precise calculations, but what were they? Do the calculations of the Commentatio have anything to do with this lecture? Or could Riemann have conceived them in a purely algebraic or analytic manner?

In the absence of relevant manuscript materials, the answer to these questions can only be conjectural. The best we can do is to examine the resources available to Riemann and to guess how he might have exploited them. Most important in this regard are Gauss's Disquisitiones of 1828 on curved surfaces, for Riemann himself regarded his concept of curvature as a generalization of the intrinsic curvature invented by Gauss in the two-dimensional case. This is why the first section of this essay is devoted to Gauss's theory of surfaces and to a rephrasing of his main results and proofs in a manner that lends itself to higherdimensional generalization. I have thereby tried to avoid notions that would not have been available to Riemann.

In the second section, I use this Gaussian background to analyze Riemann's results regarding the curvature of a manifold in his habilitation lecture on the one hand, and his results regarding the transformation properties of quadratic differential forms in the Commentatio on the other hand (a quadratic differential form is an expression of the type

\sum_{μ ν} a_{μ ν} (x) d x^{μ} d x^{ν})

. The former results concern the distortion of geodesic triangles and what is now called sectional curvature, and they involve a system of quantities which correspond to the components of the modern Riemann tensor and which I will henceforth call the Riemann parenthesis in order to avoid anachronism. The latter results consist in a criterion for the existence of a coordinate transformation that turns a given quadratic differential form into a quadratic differential form with constant coefficients; in this criterion we now recognize the vanishing of the Riemann tensor associated with the quadratic differential form. The two main issues are whether the generalization of Gaussian notions naturally leads to Riemann's concepts of geodetic distortion and sectional curvature, and whether the calculations of the Commentatio could be done by purely algebraic means, without appeal to geometric intuition. The first issue can only be addressed in a conjectural manner, as long as no relevant manuscript sources are known. For the second issue, a few relevant sheets in Riemann's hand can be found in the Göttingen University archive. The comparison of my reconstructions with Riemann's private calculations confirms the conjectures I made before seeing the archival materials.

The third section is about the way Riemann's first readers exploited the geometric and analytic-algebraic aspects of Riemannian curvature, be it in attempts to fill the gaps in Riemann's texts, or in the development of theories that formally imply this notion. The symbolic expression of this notion is the Riemann parenthesis, which has a geometrical aspect as the measure of curvature in a metric manifold, and an algebraic aspect as the crucial quantity in the equivalence problem for quadratic differential forms. From a modern point of view these two aspects are conflated because tensor analysis is usually introduced in the context of differential manifolds. Although Riemann clearly perceived the interconnection between the two aspects, he also understood that they could be developed independently: the mostly algebraic nature of the relevant section of the Commentatio is evidence for that. Some of his followers, including Dedekind, Tullio Levi-Civita, and Hermann Weyl, combined both aspects. Others, including Elwin Bruno Christoffel, Rudolf Lipschitz, and Gregorio Ricci-Curbastro favored the algebraic or analytic aspect. Did the former exaggerate Riemann's own geometric motivation? Did the latter exaggerate the independence of the algebraic aspect? For both kinds of actors, did the mysteries of Riemann's curvature stimulate mathematical creativity?

Answers to the various questions raised in this introduction will be found in the conclusion. The main text is not meant to be a history of post-Riemannian tensor calculus or differential geometry. It is only a history of a special difficulty in understanding Riemann's contributions to these topics, based on a narrow selection of relevant authors. It may, however, contribute to a better understanding of the interplay of geometric and analytical aspects of proto-tensor calculus; it may better explain the connections of this calculus with the theory of invariants (for Christoffel and Ricci) and with mechanics (for Lipschitz); and it reveals the motivations behind Levi-Civita's introduction of a central concept of modern differential geometry, parallel transport.

In the formal developments presented in the main text, be they authentic or imagined, I have freely used modern vector and tensor notation for the sake of compactness and legibility. These notations should only be regarded as abbreviations for the Cartesian-coordinate or sum-index notations that were actually used by Gauss, Riemann, and their followers, with no intended allusion to invariance or covariance properties. However, I have not tried to modernize notation for differentials and variations (

d

and

δ

symbols), because the ambiguities inherent in this notation played a significant historical role. It will be seen that the meaning of these symbols much depended on the context of reasoning, with possible misunderstandings for instance between Riemann and Levi-Civita, and interpretive difficulties for modern readers accustomed to more precise notation.

1. Gauss's intrinsic geometry of surfaces

In the mid-1820s, the Göttingen astronomer and mathematician Carl Friedrich Gauss investigated the geometry of curved surfaces in a triple context of maps, developable surfaces, and geodesy. In an influential
memoir of 1779, Joseph Louis Lagrange had given the mathematical condition for a map to conserve angles: the mapping function should satisfy the d'Alembert-Cauchy-Riemann condition, or, equivalently, it should be the real part of a derivable function of a complex variable. A similar problem, on which Gauss had been reflecting for several years, is to find the condition for a map to preserve the area of surface elements. This problem led Gauss to focus on the properties of a surface that were conserved by "development," that is, by bending without stretching. Through his involvement in a geodetic survey of the state of Hannover, Gauss was also concerned with what I shall call geodetic distortion, namely, the error committed in geodetic surveys when geodesic triangles on the curved surface of the earth are assimilated with planar triangles. As Adrien-Marie Legendre had noted in 1787, this error depends on the excess of the sum of the angles over a flat angle (Legendre, 1787, 358). As was known since the sixteenth century from Albert Girard, this excess is equal to the area of the triangle divided by the square of the radius of the sphere. Equivalently, the inverse square of the radius is the angular excess divided by the area of the triangle. Gauss presumably exploited this result to define the curvature at a point of an arbitrary curved surface by the ratio of excess angle over area for a small geodesic triangle around this point. As this definition depends only on intrinsic metric relations, the resulting expression of the curvature is invariant by development of the surface.

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Let us develop the consequences of this definition in modern vector notation, the vectors referring to the three-dimensional Euclidean space in which the surface is immersed. Call dr the vector joining two infinitesimally close points of the surface, and

d s

the length of this vector. By definition, a geodesic line is a line of minimal length, for which

δ \int d s = 0

for fixed extremities of the line. Calling

u

the tangent unit vector

d r / d s

along the geodesic, and

n

the unit vector normal to the surface, the resulting equation of a geodesic reads

\frac{d u}{d s} = - (u \cdot \frac{d n}{d s}) n = u \times (\frac{d n}{d s} \times n)

This means that during a displacement

d s

along a geodesic, the tangent unit vector rotates by the vector angle

d n \times n

. More generally, a unit vector a making a constant angle with the geodesic rotates by the same amount, so that

\begin{aligned} \frac{d a}{d s} = - (a \cdot \frac{d n}{d s}) n, and \\ \frac{d^{2} a}{d s^{2}} = - (a \cdot \frac{d^{2} n}{d s^{2}}) n - (a \cdot \frac{d n}{d s}) \frac{d n}{d s} . \end{aligned}

On the geodetic line connecting the points

r

and

r + \hat{d} r

, and to second order in

\hat{d} s

, the vector

a

changes by

\hat{d a} = \frac{d a}{d s} \hat{d} s + \frac{1}{2} \frac{d^{2} a}{d s^{2}} (\hat{d} s)^{2} = - (a \cdot \hat{d} n) (n + \frac{1}{2} \hat{d} n)

Now consider a small geodesic triangle with summits at

r, r + \hat{d} r

and

r + \hat{δ} r

. Call

u

the tangent unit vector at point

r

of the geodesic joining

r

and

r + \hat{d} r; u^{'}

the tangent unit vector at point

r + \hat{d} r

of the same geodesic;

Figure 1. Angles of a small geodesic triangle.

u^{''}

the unit vector that makes the same angle with the geodesic joining

r + \hat{d} r

and

r + \hat{δ} r

at point

r + \hat{δ} r

as the vector

u^{'}

does with the same geodesic at point

r + \hat{d} r

; and

v

the unit vector that makes the same angle with the geodesic joining

r

and

r + \hat{δ} r

at point

r + \hat{δ} r

as the vector

u

does with the same geodesic at point

u

(see Figure 1). By construction, the angle between

u^{''}

and

- v

is equal to the sum of the angles of the geodesic triangle. Equivalently, the angle between

u^{''}

and

v

is the excess of this sum over a flat angle. To second order with respect to the sides of the triangle, we may compute

u^{'}, u^{''}

, and

v

by rule (4). This leads to

u^{''} - v = \frac{1}{2} u \times (\hat{δ} n \times \hat{d} n)

Consequently, the sum of the angles of a small geodesic triangle differs from a flat angle by an amount given by the area of the triangle defined by the vectors

\hat{d} n

and

\hat{δ} n

on the unit sphere

n^{2} = 1 .^{5}

The Gaussian curvature

C

, if it is defined as the ratio of excess angle over area, should therefore satisfy

\hat{d} n \times \hat{δ} n = C \hat{d} r \times \hat{δ} r

The consistency of this definition requires that the coefficient

C

should be the same for every triangle. Gauss proved this by relating

C

to Euler's principal curvatures in the following manner. At a given point of the surface, we may choose a Cartesian system of coordinates with the origin at this point and the

z

-axis normal to the surface at this point. In these coordinates, the equation of the surface takes the form

z = A x^{2} + B x y + C y^{2} + \dots

. For a proper choice

(X, Y)

of rectangular coordinates in the tangent plane, we have the diagonal form

z = X^{2} / 2 ρ \pm Y^{2} / 2 σ + \dots

. Any plane containing the normal direction at the origin intersects the surface on a line whose curvature varies between the extreme values

1 / ρ

and

1 / σ

, which are reached when the plane contains the

X

and

Y

axes respectively. These are the principal curvatures introduced by Leonhard Euler in (1767). To first order in the distance from the origin, the normal unit vector is

n = (X / ρ, \pm Y / σ, - 1)

, whence follows

d n \times δ n = (0, 0, \pm 1 / ρ σ) (d X δ Y - d Y δ X) = C (d r \times δ r), with C = \pm 1 / ρ σ

for every triangle defined by the vectors

d r = (d X, d Y, 0)

and

δ r = (δ X, δ Y, 0)

The Gaussian curvature is therefore equal to the product of the two principal curvatures, with a positive sign in the convexo-convex case, and a negative sign in the convexo-concave case.

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Plausibly, Gauss's discovery of a relation between the angular excess of a geodesic triangle and the image of this triangle on the

n^{2} = 1

sphere led him to redefine the intrinsic curvature through the variations of the normal unit vector. For a small portion of the surface, the normal unit vectors corresponding to points of this portion describe a small portion of the unit sphere. In Gauss's Disquisitiones of 1828, the curvature is defined as the ratio of the corresponding areas (Gauss,

1828, § 6

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In an early draft of this memoir (1825, §16), Gauss proved the intrinsic character of this definition by relating it to the angular excess of geodesic triangles, as we just did. In the final version, he followed a more algebraic route in which the curvature is first expressed in terms of the parametric representation

r (p, q)

of the surface and its derivatives and then shown to depend only on the coefficients of the fundamental form

d s^{2} = E d p^{2} + 2 F d p d q + G d q^{2}

that gives the element of length in terms of the variations of the parameters

p

and

q

This is the theorema egregium, which Gauss stated as follows

(1828, § 12)

The analysis developed in the preceding article shows us that for finding the measure of curvature there is no need of finite formulæ that express the coordinates

x, y, z

as functions of the indeterminates

p, q

; but that the general expression for the magnitude of any linear element is sufficient.

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As the expression of the element of length evidently does not change by development of the surface, Gauss's theorem implies the intrinsic character of the Gaussian curvature.

The expression of the curvature in terms of the

E, F, G

coefficients being in itself an important result, I will derive it in a simpler way than Gauss's, by directly exploiting the geodetic definition of the Gaussian curvature. In order to prepare later generalization, I use the indicial expression of the metric

d s^{2} = g_{μ ν} d x^{μ} d x^{ν}

, wherein the indices take the values 1 and 2 , with

x_{1} = p, x_{2} = q, g_{11} = E

g_{12} = g_{21} = F, g_{22} = G

, and Einstein's summation rule is used. If

r (x_{1}, x_{2})

denotes the point of the surface corresponding to the values

x_{1}

and

x_{2}

of the parameters, then

g_{μ ν} = \partial_{μ} r \cdot \partial_{ν} r

. The equation of a geodesic line results from

0 = δ \int \sqrt{g_{μ ν} d x^{μ} d x^{ν}} = \int \frac{1}{2} \partial_{ρ} g_{μ ν} δ x^{ρ} d x^{μ} d x^{ν} / d s + \int g_{μ ν} (d x^{μ} / d s) δ d x^{ν}

Introducing

u^{μ} = d x^{μ} / d s

, and integrating the last integral by parts, we get

\begin{aligned} \frac{d u^{ρ}}{d s} + Γ_{μ ν}^{ρ} u^{μ} u^{ν} = 0, with \\ g_{ρ σ} Γ_{μ ν}^{ρ} = \frac{1}{2} (\partial_{μ} g_{σ ν} + \partial_{ν} g_{σ μ} - \partial_{σ} g_{μ ν}) \end{aligned}

The couple

(u^{1}, u^{2})

gives the coordinates of the unit tangent vector

u

in the tangent-plane basis

(\partial_{1} r, \partial_{2} r)

. Similarly, any vector

a

in the tangent plane can be expressed by its coordinates

a^{μ}

in this local basis. The scalar product of two vectors

a

and

b

is then given by

a \cdot b = a^{μ} b^{ν} \partial_{μ} r \cdot \partial_{ν} r = g_{μ ν} a^{μ} b^{ν}

In order to imitate the earlier reasoning on a small geodesic triangle, consider a unit vector a that makes a constant angle with the vector

u

along the geodesic.

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The coordinates of this vector evolve according to

\frac{d a^{ρ}}{d s} + Γ_{μ ν}^{ρ} u^{μ} a^{v} = 0

because this equation is easily seen to preserve

a^{2} = g_{μ ν} a^{μ} a^{ν}

and

a \cdot u = g_{μ ν} a^{μ} u^{ν}

In an alleviated notation, the former equation reads

d a = - Γ_{a, d x}

. A second differentiation along the geodesic leads to

d^{2} a = - d Γ_{a, d x} - Γ_{d a, d x} - Γ_{a, d^{2} x}, with d Γ = d x^{μ} \partial_{μ} Γ .

Hence follows

\hat{d} a = - Γ_{a, \hat{d} x} - \frac{1}{2} Γ_{\hat{d} a, \hat{d} x} - \frac{1}{2} \hat{d} Γ_{a, \hat{d} x}

for the variation of

a

on the geodesic joining

x

and

x + \hat{d} x

to second order in

\hat{d} x

. We may now compute the vectors

u^{'}, u^{''}

, and

v

of Figure 1 through this formula. Writing

d

for

\hat{d}

, and

δ

for

\hat{δ}

, this leads to

u^{''} - v = - \frac{1}{2} (Γ_{d u, δ x} - Γ_{δ u, d x} + d Γ_{u, δ x} - δ Γ_{u, d x})

Returning to the tensor notation, this is

\begin{aligned} u^{'' μ} - v^{μ} = - \frac{1}{2} R^{μ}_{v ρ σ} u^{ν} d x^{ρ} δ x^{σ}, with \\ R^{μ}_{ν ρ σ} = \partial_{ρ} Γ_{ν σ}^{μ} - \partial_{σ} Γ_{ν ρ}^{μ} + Γ_{ρ τ}^{μ} Γ_{ν σ}^{τ} - Γ_{σ τ}^{μ} Γ_{ν ρ}^{τ} . \end{aligned}

Incidentally, this reasoning shows that for any three vectors

a, b

, and

c

, the expression

R^{μ}_{ν ρ σ} a^{ν} b^{ρ} c^{σ}

transforms like the coordinates of a vector under a change of coordinates.

In order to compare formula (16) with the analogous extrinsic formula (5), we form the scalar product

(u^{''} - v) \cdot w = g_{μ ν} (u^{'' μ} - v^{μ}) w^{ν}

. Using the definition (6) of the curvature, we get

(u^{''} - v) \cdot w = \frac{1}{2} C (u \times w) \cdot (d r \times δ r) = \frac{1}{2} C u^{μ} w^{ν} d x^{ρ} δ x^{σ} (g_{μ ρ} g_{ν σ} - g_{μ σ} g_{ν ρ}),

to be compared with

g_{μ ν} (u^{'' μ} - v^{μ}) w^{ν} = - \frac{1}{2} R_{μ ν ρ σ} w^{μ} u^{ν} d x^{ρ} δ x^{σ}, with R_{μ ν ρ σ} = g_{μ τ} R^{τ}_{ν ρ σ}

We therefore have

R_{μ ν ρ σ} = C (g_{μ ρ} g_{v σ} - g_{μ σ} g_{v ρ}) .

This implies the symmetry relations

R_{μ ν ρ σ} = R_{ρ σ μ ν}, R_{μ ν σ ρ} = - R_{μ ν ρ σ}, R_{ν μ ρ σ} = - R_{μ ν ρ σ},

and there is only one independent

R_{μ ν ρ σ}

coefficient, say

R_{1212} = C (g_{11} g_{22} - g_{12}^{2})

. Through this formula and through Eqs. (11) and (17), the curvature

C

is seen to depend only on the

g_{μ ν}

coefficients and their first and second derivatives, as was to be proved.

So far we have encountered two interpretations of curvature in Gauss's theory of surfaces: through the angular excess of geodesic triangles, and through the surface swept by the normal unit vector on the unit sphere. The last section of Gauss's memoir (1828, §§23-29) provides a third interpretation through geodetic distortion, which is the surveying error committed in assimilating geodesic triangles with planar triangles. As was already hinted to, Legendre

(1787, 358)

proved that the ratios between the sides of a small spherical triangle of angles

α, β

, and

γ

, were nearly the same as for a planar triangle of angles

α - e / 3, β - e / 3

γ - e / 3

, wherein

e

is the excess of the sum of the angles of the spherical triangle over the flat angle

π

. Gauss generalized this result to surfaces of variable curvature and obtained a higher-order correction for the large triangles he had to consider in his geodetic mission. Calling

a, b, c

the sides of the geodesic triangle, the lowest-order correction yields

a^{2} = b^{2} + c^{2} - 2 b c \cos (α - e / 3) \approx b^{2} + c^{2} - 2 b c \cos α - (2 / 3) e b c \sin α .

Calling

S

the area of the triangle, and remembering that

C = e / S

, the correction to planar geodesy is

^{10}

a^{2} - (b^{2} + c^{2} - 2 b c \cos α) = - (4 / 3) C S^{2} .

Following Gauss, we may express this result in geodesic polar coordinates generalizing the latitude and longitude on a sphere. For a given point of the surface (not too far from the pole), the first of these coordinates is the geodesic distance

s

from the pole, and the second is the angle

θ

that the connecting geodesic makes with a fixed direction at the pole. In these coordinates and in a first approximation, the expression of the distance between two points differs from the expression it would take on a plane surface by

- (4 / 3) C S^{2}

It will be useful to prove this result by Gauss in a way that easily extends to higher dimensions. From the geodesic polar coordinates

s

and

θ

, we build the normal coordinates

x^{1} = s \cos θ

and

x^{2} = s \sin θ

, alias

x^{μ} = s ξ^{μ}

. Along a geodesic from the pole,

\frac{d x^{μ}}{d s} = ξ^{μ}, Γ_{ν ρ}^{μ} ξ^{v} ξ^{ρ} = - \frac{d^{2} x^{μ}}{d s^{2}} = 0, and ξ^{σ} \partial_{σ} Γ_{ν ρ}^{μ} ξ^{ν} ξ^{ρ} = - \frac{d^{3} x^{μ}}{d s^{3}} = 0

The direction of the

ξ

vector being arbitrary, the second identity requires the

Γ

coefficients to vanish. The third identity requires

\partial_{σ} Γ_{ν ρ}^{μ} + \partial_{ν} Γ_{ρ σ}^{μ} + \partial_{ρ} Γ_{σ ν}^{μ} = 0 .

Now consider the infinitesimal vector

d x

joining the points of coordinates

δ x

and

δ x + d x

, and the vector

d x^{'}

obtained by moving this vector to the pole along the geodesic joining the point of coordinates

δ x

to the pole in the length-preserving manner given by formula (14). Since in the normal coordinates the

Γ

coefficients vanish, we have

d x^{'} = d x + \frac{1}{2} δ Γ_{d x, δ x}

. The length of this vector being the same as the length

d s^{2'}

of the vector

d x

, we have (to fourth order)

d s^{2'} = d s^{2} + d x \cdot δ Γ_{d x, δ x}

, wherein

d s^{2}

is the expression that the length would have in the planar case, and the dot product is taken with respect to the

g_{μ ν}

at the pole. Identity (25) and the expression (17) of

R^{μ}_{ν ρ σ}

imply

2 δ Γ_{d x, δ x} + d Γ_{δ x, δ x} = 0, and δ Γ_{d x, δ x} = \frac{1}{3} (δ Γ_{d x, δ x} - d Γ_{δ x, δ x}) = - \frac{1}{3} δ x^{ν} R^{μ}_{ν ρ σ} d x^{ρ} δ x^{σ}

so that the correction to the metric is

δ d s^{2} = - \frac{1}{3} R_{μ ν ρ σ} d x^{μ} δ x^{ν} d x^{ρ} δ x^{σ}

In passing we may note that owing to the symmetries (21) of

R_{μ v ρ σ}

, this expression is a quadratic form of the antisymmetric combinations

d x^{μ} δ x^{ν} - δ x^{μ} d x^{ν}

(and therefore vanishes when

d x

and

δ x

are parallel, as expected on geometrical grounds). Using the two-dimensional expression (20) of

R_{μ ν ρ σ}

, we finally get the distortion

δ d s^{2} = - \frac{1}{3} C [(d r)^{2} (δ r)^{2} - (d r \cdot δ r)^{2}] = - (4 / 3) C S^{2},

wherein

S

is the surface of the triangle defined by the pole and the points of coordinates

δ x

and

δ x + d x .^{11}

2. Riemann's curvature

In an abstract of his Disquisitiones (1828, 45), Gauss wrote:

These theorems lead to the consideration of the theory of curved surfaces from a new point of view, where a wide and still wholly uncultivated field is open to investigation. If we consider surfaces not as boundaries of bodies, but as bodies of which one dimension vanishes, and if at the same time we conceive them as flexible but not extensible, we see that two essentially different relations must be distinguished, namely, on the one hand, those that presuppose a definite form of the surface in space; on the other hand, those that are independent of the various forms that the surface may assume. This discussion is concerned with the latter. In accordance with what has been said, the measure of curvature belongs to this case. But it is easily seen that the consideration of figures constructed upon the surface, their angles, their areas and their integral curvatures, the joining of the points by means of shortest lines, and the like, also belong to this case. All such investigations must start from this, that the very nature of the curved surface is given by means of the expression of any linear element in the form

\sqrt{E d p^{2} + 2 F d p d q + G d q^{2}}^{12}

Gauss's interpretation of surfaces as "bodies of which one dimension vanishes" and his focus on the intrinsic geometry of such surfaces invited generalization to manifolds of any dimension.

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2.1. Riemann's habilitation lecture

Gauss's student Bernhard Riemann got the hint and developed it in his habilitation lecture of 1854 (Riemann, 1867), in a manner that Gauss himself judged highly impressive. Riemann first defined a continuous, differentiable manifold of

n

dimensions, in an intuitive manner that need not concern us here. For the purpose of length measurement, he then defined a homogenous differential form

^{14}

on this manifold, focusing on the Gaussian quadratic case that we would now write as

d s^{2} = g_{μ ν} d x^{μ} d x^{ν}

Before proceeding to the analysis of this metric structure, Riemann made clear that Gauss's memoir on surfaces would be his guide (Riemann, 1873, 15):

These measure-relations can only be studied in abstract notions of quantity, and their dependence on one another can only be represented by formulae. On certain assumptions, however, they are decomposable into relations which, taken separately, are capable of geometric representation; and thus it becomes possible to express geometrically the calculated results. In this way, to come to solid ground, we cannot, it is true, avoid abstract considerations in our formulae, but at least the results of calculation may subsequently be presented in a geometric form. The foundations of these two parts of the question are established in the celebrated memoir of Gauss, Disquisitiones generales circa superficies curvas.

^{15}

From this extract, it is clear that Riemann performed detailed calculations generalizing Gauss's calculation of intrinsic surface properties, and that he also found a way to extend Gauss's geometric interpretation of these properties. As was mentioned, in his lecture Riemann only gave the results of the calculations and their interpretation.

^{16}

The coefficients

g_{μ ν}

depend both on the metric structure of the manifold and on the arbitrary choice of coordinates. In order to avoid this indeterminacy, Riemann

(1867, 141)

introduced special coordinates that could be defined though intrinsic metric operations. If

O

is a given point (pole) of the manifold, the position of another point

M

(in the neighborhood of

O

) can be determined by drawing the geodesic from

O

M

and recording the length

s

of this geodesic as well as the rectangular coordinates

ξ^{μ}

of the unit tangent vector at the beginning of this geodesic. This is a straightforward generalization of Gauss's geodesic polar coordinates. Riemann next defines the so-called geodesic normal coordinates through

x^{μ} = s ξ^{μ}

, as a natural generalization of Cartesian coordinates to curved spaces.

^{17}

In the flat case, all geodesics are straight lines and the metric element is simply given by

d s^{2} = {(d x^{1})}^{2} + {(d x^{2})}^{2} + \dots + {(d x^{n})}^{2}

throughout space. In the general case, Riemann asserts that at the lowest order in

x

the metric element differs from its Euclidean value by a quadratic form in the combinations

x^{μ} d x^{ν} - x^{ν} d x^{μ}

. Formally, this statement may be rendered as

δ d s^{2} = \frac{1}{4} S_{μ ν ρ σ} (x^{μ} d x^{ν} - x^{ν} d x^{μ}) (x^{ρ} d x^{σ} - x^{σ} d x^{ρ})

with

S_{μ ν ρ σ} = S_{ρ σ μ ν}, S_{μ ν σ ρ} = - S_{μ ν ρ σ}, S_{ν μ ρ σ} = - S_{μ ν ρ σ}

Riemann

(1867, 142 - 143)

further notes that the ratio of this quantity to the square of the area of the triangle of vertices

O, x

, and

x + d x

(which is

1 / 4

x^{2} (d x)^{2} - (x \cdot d x)^{2}

in a first approximation) is a finite quantity independent of the choices of

x

and

d x

as long as the associated triangle (more exactly the O edge of this triangle) remains in the same tangent plane. Lastly, Riemann tells us that the latter quantity multiplied by

- 3 / 4

gives the Gaussian curvature of the surface defined by the set of geodesics whose initial tangent vector belongs to the tangent plane associated with a given value of

x

and

d x

. In symbols, and writing

δ x

instead of

x

(for the sake of conformity with our earlier notation), this curvature is

C (d x, δ x) = - 3 \frac{S_{μ ν ρ σ} d x^{μ} δ x^{ν} d x^{ρ} δ x^{σ}}{(g_{μ ν} d x^{μ} d x^{ν}) (g_{ρ σ} δ x^{ρ} δ x^{σ}) - {(g_{μ ν} d x^{μ} δ x^{ν})}^{2}}

Riemann never published a proof of these statements. Plausibly, he reasoned in the manner given above in the two-dimensional case,

^{18}

or according to the more complex variants proposed in (Dedekind and Weber, 1876) and (Weyl, 1919). Such reasoning leads, for any dimension, to Eq. (27)

δ d s^{2} = - \frac{1}{3} R_{μ v ρ σ} d x^{μ} δ x^{ν} d x^{ρ} δ x^{σ}

with the expression (17) of

R_{μ ν ρ σ}

. In order to reach Riemann's quadratic expression (29) of the geodetic distortion, one further needs the symmetry relations (21):

R_{μ ν ρ σ} = R_{ρ σ μ ν}, R_{μ ν σ ρ} = - R_{μ ν ρ σ}, R_{ν μ ρ σ} = - R_{μ ν ρ σ} .

The second relation is the only one that can immediately be read in the expression of

R_{μ ν ρ σ}

. Although the other ones can be derived from this expression, they do not jump to the eyes. Plausibly, Riemann first noted them in the two dimensional-case in the manner given above, and then checked their validity for any dimension.

By definition, geodetic distortion derives from the metric relations of the manifold. Reciprocally, the curvature coefficients expressing this distortion completely determine the metric relations as long as these coefficients are independent from each other. Riemann

(1867, 140, 142)

justified this proposition by arguing that there were

n (n - 1) / 2

independent curvature coefficients in general (as many as there are independent surface directions

δ x^{μ} d x^{ν} - δ x^{ν} d x^{μ}

) and just as many independent changes of the

g_{μ ν}

coefficients that could not be absorbed by a change of coordinates (as there are

n (n + 1) / 2

metric coefficients and

n

coordinates). He may also have relied on the intuitive remark that the knowledge of the distortion of small geodesic triangles is sufficient to correctly survey the whole manifold. Consequently, a metric manifold of everywhere vanishing curvature is an Euclidean space (Riemann, 1867, 144).

^{19}

As for the relation (30) between geodetic distortion and the Gaussian curvature of the geodetic surface tangent to the plane of the vectors

d x

and

δ x

, Riemann is likely to have obtained it by direct calculation in normal geodesic coordinates for which this surface is defined by

x_{3} = x_{4} = \dots = x_{n} = 0 .^{20}

Indeed in such coordinates, we can rely on the above-given two-dimensional derivation of the relation between the angular excess of geodesic triangles and geodetic distortion.

2.2. Riemann's Commentatio

In 1861 Riemann sent a manuscript to the French Academy of Science (Riemann, 1876, 370-383) in answer to a prize question on the propagation of heat in a homogenous (and isotropic) solid: find the conditions under which the temperature becomes a function of time and of two independent (spatial) variables only. In symbols, this means that the temperature

θ

is a function of time and of two functions

λ (r)

and

μ (r)

of the position

r

in the solid. Consequently, the lines formed by the intersections of the surfaces of constant

λ

and constant

μ

are isothermal lines at any time. In order to solve this problem, Riemann introduced curvilinear coordinates

(s_{1}, s_{2}, s_{3})

such that

s_{1} = λ

and

s_{2} = μ

. The equation of propagation of heat being

h \frac{\partial θ}{\partial t} = Δ θ = \frac{\partial^{2} θ}{\partial x^{2}} + \frac{\partial^{2} θ}{\partial y^{2}} + \frac{\partial^{2} θ}{\partial z^{2}}

in Cartesian coordinates, it takes the form

K \frac{\partial θ}{\partial t} = \sum_{i j} \partial_{i} (b_{i j} \partial_{j} θ)

in the new system of coordinates. The coefficients

K

and

b_{i j}

are now functions of the

s_{i}

coordinates as they would be in a heterogeneous anisotropic solid. Riemann's strategy was first to determine for which choices of the

b_{i j}

coefficients the temperature in such a solid would depend on the two coordinates

s_{1}

and

s_{2}

only, and then to further require these coefficients to be such that the equation of propagation could be brought to the isotropic homogenous form (31) by a change of variables.

^{21}

Calling

β_{i j}

the matrix of minor determinants of the matrix

b_{i j}

, the latter condition is equivalent to the existence of a system

(x, y, z)

of coordinates for which the form

β_{i j} d s_{i} d s_{j}

becomes

d x^{2} + d y^{2} + d z^{2} .^{22}

Riemann was thus brought to the following question: Under what conditions on the

g_{μ ν}

coefficients can the form

g_{μ ν} (x) d x^{μ} d x^{ν}

be brought to the simple form

d r^{2}

through the change of variables

x \to r (x)

? Toward the beginning of his memoir (p. 372), Riemann announced: "We will see below that this question can be handled by nearly the same method that Gauss used in the theory of curved surfaces." 23 This handling is found in the second part of the memoir (pp. 380-383). I now proceed to describe this part without prejudging the importance of the Gaussian background.

As Riemann first shows, the transformability problem is equivalent to the problem of finding under which condition the system of partial differential equations

g_{μ ν} = \partial_{μ} r \cdot \partial_{ν} r

admits a solution. Riemann's next step is to replace this non-linear system of first differential order with a linear system of second differential order. For this purpose, he differentiates the former equation to get

\partial_{ρ} g_{μ ν} = \partial_{ρ} \partial_{μ} r \cdot \partial_{ν} r + \partial_{μ} r \cdot \partial_{ρ} \partial_{ν} r .

Subtracting from this equation the two equations obtained by circular permutation of the indices, he then gets

^{24}

\partial_{μ} \partial_{ν} r \cdot \partial_{ρ} r = Γ_{μ ν ρ}, with Γ_{μ v ρ} = \frac{1}{2} (\partial_{ν} g_{μ ρ} + \partial_{ρ} g_{μ ν} - \partial_{μ} g_{ν ρ}) .

Using Eq. (33), this is equivalent to

\partial_{μ} \partial_{ν} r = Γ_{μ ν}^{ρ} \partial_{ρ} r

For an unknown function

f

, the compatibility of the equations of the system

\partial_{μ} f = F_{μ}

requires

\partial_{μ} F_{ν} -

\partial_{ν} F_{μ} = 0

. Riemann similarly requires

\partial_{σ} (Γ_{μ ν}^{ρ} \partial_{ρ} r) - \partial_{μ} (Γ_{σ ν}^{ρ} \partial_{ρ} r) = 0

Together with Eq. (35), this condition implies

R^{μ}_{v ρ σ} = 0, with R^{μ}_{v ρ σ} = \partial_{ρ} Γ_{v σ}^{μ} - \partial_{σ} Γ_{v ρ}^{μ} + Γ_{ρ τ}^{μ} Γ_{v σ}^{τ} - Γ_{σ τ}^{μ} Γ_{v ρ}^{τ} .

In Riemann's notation, this is

(ι^{'}, ι^{''} ι^{'''}) = 0

, with

(ι^{'}, ι^{''} ι^{'''}) = - 2 R_{ι^{'} ι^{''} ι^{'''}}

In this paragraph, Riemann does not make any reference to the evident geometric interpretation of his problem. He only gives a series of algebraic manipulations. Does this mean that he privately arrived at condition (38) by purely algebraic means? This is doubtful, because the move from Eq. (34) to Eq. (36) seems farfetched from an algebraic point of view. More likely, Riemann reasoned that in a manifold with the metric

g_{μ ν}

, the geodesics should be the same in both systems of coordinates. In symbols,

{\ddot{x}}^{μ} + Γ_{v ρ}^{μ} {\dot{x}}^{v} {\dot{x}}^{ρ} = 0 \Leftrightarrow \ddot{r} = 0

wherein the dot derivative is taken with respect to the curvilinear abscissa along the geodesic. Using

r =

{\dot{x}}^{μ} \partial_{μ} r

and

\ddot{r} = {\ddot{x}}^{μ} \partial_{μ} r + {\dot{x}}^{μ} {\dot{x}}^{ν} \partial_{μ} \partial_{ν} r

, we must have

{\dot{x}}^{μ} {\dot{x}}^{ν} \partial_{μ} \partial_{ν} r = Γ_{ν ρ}^{μ} {\dot{x}}^{ν} {\dot{x}}^{ρ} \partial_{μ} r

. Since the derivatives

{\dot{x}}^{μ}

can take any value, and since

\partial_{μ} \partial_{ν} r

and

Γ_{μ ν}^{ρ} \partial_{ρ} r

are both symmetric with respect to a permutation of the indices

μ

and

v

, this requires the validity of Eq. (36).

^{25}

The next paragraph of Riemann's Commentatio is most mysterious. There, "in order to reveal the character of the equations

{[R_{μ ν ρ σ} = 0]}^{" 26}

he forms the expression

X = δ δ (g_{μ ν} d x^{μ} d x^{ν}) - 2 d δ (g_{μ ν} d x^{μ} δ x^{ν}) + dd (g_{μ ν} δ x^{μ} δ x^{ν})

in which the second-order variations

d^{2}, δ^{2}

, and

d δ

are determined so that

\begin{aligned} δ^{'} (g_{μ ν} d x^{μ} d x^{ν}) - 2 d (g_{μ ν} d x^{μ} δ^{'} x^{ν}) = 0, δ^{'} (g_{μ ν} δ x^{μ} δ x^{ν}) - 2 δ (g_{μ ν} δ x^{μ} δ^{'} x^{ν}) = 0, \\ δ^{'} (g_{μ ν} d x^{μ} δ x^{ν}) - δ (g_{μ ν} d x^{μ} δ^{'} x^{ν}) - d (g_{μ ν} δ x^{μ} δ^{'} x^{ν}) = 0 \end{aligned}

for any variation

δ^{'}

. Without proof, Riemann announces

^{27}

X = - \frac{1}{2} R_{μ ν ρ σ} (d x^{μ} δ x^{ν} - d x^{ν} δ x^{μ}) (d x^{ρ} δ x^{σ} - d x^{σ} δ x^{ρ})

with the comment

(1876, 382)

: "From the buildup of this expression, it is clear that through a change of the independent variables this expression is turned into an expression that depends on the new form of

g_{μ ν} d x^{μ} d x^{ν}

by the same law."

^{28}

Riemann probably meant that the quantity

X

, which is an invariant by construction, is the same combination of

R_{μ ν ρ σ}, d x^{μ}

and

δ x^{μ}

whatever be the choice of the coordinates

x^{μ}

. Taking into account the vector character of

d x^{μ}

and

δ x^{μ}

and the symmetries of

R_{μ ν ρ σ}

by permutation of the indices, this remark is tantamount to recognizing the tensor character of

R_{μ ν ρ σ}

Riemann goes on constructing the quantity

Y = \frac{R_{μ ν ρ σ} (d x^{μ} δ x^{ν} - d x^{ν} δ x^{μ}) (d x^{ρ} δ x^{σ} - d x^{σ} δ x^{ρ})}{4 [(g_{μ ν} d x^{μ} d x^{ν}) (g_{ρ σ} δ x^{ρ} δ x^{σ}) - {(g_{μ ν} d x^{μ} δ x^{ν})}^{2}]}

which is evidently meant to relate

R_{μ ν ρ σ}

to the curvature introduced in his lecture of 1854. In this context, he knew that in geodesic normal coordinates

Y

represented the Gaussian curvature

C (d x, δ x)

in the "surface direction" defined by the vectors

d x

and

δ x

and that it therefore did not depend on the choice of these vectors as long as they belonged to the same tangent plane (compare with Eq. (30), for

S_{μ ν ρ σ} = - \frac{1}{3} R_{μ v ρ σ}

). The

X

and

Y

quantities being invariant, this interpretation is preserved under a change of coordinates. The vanishing of

Y

in any given system of coordinates therefore implies the vanishing of the curvature of the associated metric manifold and the existence of a system of coordinates leading to the uniform metric of an Euclidean space. Riemann thus knew that the vanishing of

R_{μ ν ρ σ}

, which he had proved to be a necessary condition by algebraic means, also was a sufficient condition. Riemann made these points at the very end of his memoir, again without proof. The connection with Gaussian curvature came last, after Eq. (43). It was introduced in these words

(1876, 382)^{29}

: "This investigation can be illustrated by a somewhat geometrical interpretation. Although this interpretation relies on unusual concepts, some words about it should help." 30

Although there is no doubt that Riemann introduced the quantity

Y

for geometrical reasons, it is not clear whether his introduction of the quantity

X

was of algebraic or of geometric purport. Let us first try our best to reconstruct an algebraic route to

X

. Having reached the condition

R_{μ ν ρ σ} = 0

for the uniformization of a quadratic differential form, Riemann may have wanted to know how the

R_{μ ν ρ σ}

quantities transformed under a change of coordinates. From the fact that these quantities involve second order partial derivatives of the

g_{μ ν}

coefficients, he may have suspected that they could be obtained by second-order variations of the invariants

g_{μ ν} d x^{μ} d x^{ν}, g_{μ ν} d x^{μ} δ x^{ν}

, and

g_{μ ν} δ x^{μ} δ x^{ν}

. For instance, the variation

x^{μ} \to x^{μ} + δ x^{μ}

gives

\begin{aligned} δ (g_{μ ν} d x^{μ} d x^{ν}) = & δ x^{ρ} d x^{μ} d x^{ν} \partial_{ρ} g_{μ ν} + 2 g_{μ ν} d δ x^{μ} d x^{ν}, \\ δ δ (g_{μ ν} d x^{μ} d x^{ν}) = & δ x^{ρ} δ x^{σ} d x^{μ} d x^{ν} \partial_{σ} \partial_{ρ} g_{μ ν} + δ δ x^{ρ} d x^{μ} d x^{ν} \partial_{ρ} g_{μ ν} \\ + 3 δ x^{ρ} d δ x^{μ} d x^{ν} \partial_{ρ} g_{μ ν} + 2 g_{μ ν} d δ x^{μ} d δ x^{ν} + 2 g_{μ ν} d δ δ x^{μ} d x^{ν} . \end{aligned}

The combination (40) is designed so that the terms containing the third-order differentials

d δ δ x^{μ}

and

δ dd x^{μ}

cancel each other.

^{31}

In order to eliminate the second order differentials

dd x^{μ}, δ δ x^{μ}

, and

δ d x^{μ}

, we must relate them to the first order differentials in a covariant manner. For this purpose we may take

dd x^{μ} = - Γ_{ν ρ}^{μ} d x^{ν} d x^{ρ}, δ δ x^{μ} = - Γ_{ν ρ}^{μ} δ x^{ν} δ x^{ρ}, and δ d x^{μ} = - Γ_{ν ρ}^{μ} δ x^{ν} d x^{ρ}

for the value

x^{μ}

of the coordinates of the point at which we are evaluating

X

(not for other values in the neighborhood of this value). The form of these relations is indeed preserved by a change of coordinates, as is seen from the invariance of the variational identities

\begin{aligned} δ^{'} (g_{μ ν} d x^{μ} d x^{ν}) - 2 d (g_{μ ν} d x^{μ} δ^{'} x^{ν}) = - 2 g_{μ τ} δ^{'} x^{τ} (dd x^{μ} + Γ_{ν ρ}^{μ} d x^{ν} d x^{ρ}) \\ δ^{'} (g_{μ ν} δ x^{μ} δ x^{ν}) - 2 δ (g_{μ ν} δ x^{μ} δ^{'} x^{ν}) = - 2 g_{μ τ} δ^{'} x^{τ} (δ δ x^{μ} + Γ_{ν ρ}^{μ} δ x^{ν} δ x^{ρ}) \\ δ^{'} (g_{μ ν} d x^{μ} δ x^{ν}) - δ (g_{μ ν} d x^{μ} δ^{'} x^{ν}) - d (g_{μ ν} δ x^{μ} δ^{'} x^{ν}) = - 2 g_{μ τ} δ^{'} x^{τ} (δ d x^{μ} + Γ_{ν ρ}^{μ} δ x^{ν} d x^{ρ}) \end{aligned}

With the expressions (45) of the second order differentials, a straightforward though tedious calculation leads to

X = - 2 R_{μ ν ρ σ} d x^{μ} δ x^{ν} d x^{ρ} δ x^{σ}

, in conformity with Riemann's result (42).

^{32}

With the benefit of hindsight, this result can be reached very quickly by means of the covariant variations

D

and

Δ

corresponding to the ordinary variations

d

and

δ

. Using

D f = d f

for any scalar,

D a^{μ} = d a^{μ} +

Γ_{ν ρ}^{μ} d x^{ν} a^{ρ}

for any vector

a^{μ}, D g_{μ ν} = 0

(by Ricci's lemma), the similar properties of the

Δ

variation, and the relation

D δ x^{μ} = Δ d x^{μ}

, we have

\begin{aligned} X & = Δ Δ (g_{μ ν} d x^{μ} d x^{ν}) - 2 D Δ (g_{μ ν} d x^{μ} δ x^{ν}) + DD (g_{μ ν} δ x^{μ} δ x^{ν}) \\ = 2 g_{μ ν} d x^{μ} Δ Δ d x^{ν} - 2 g_{μ ν} d x^{μ} D Δ δ x^{ν} - 2 g_{μ ν} δ x^{μ} D Δ d x^{ν} + 2 g_{μ ν} δ x^{μ} DD δ x^{ν} \\ = 2 g_{μ ν} d x^{μ} (Δ D - D Δ) δ x^{ν} = - 2 R_{μ ν ρ σ} d x^{μ} δ x^{ν} d x^{ρ} δ x^{σ} . \end{aligned}

It is important to note that in this calculation

Dd x^{μ}, D δ x^{μ}, Δ d x^{μ}, Δ δ x^{μ}

vanish only when they are not subjected to an additional variation.

^{33}

How plausible is this algebraic route? Riemann could have been aware of a section of Lagrange's Mécanique analytique (1811-1815, 1:307-310) in which the invariance of the differential expression

f (x, d x)

(when the coordinates

x

are changed to

x^{'}

and the function

f

is changed to

f^{'}

) is shown to imply the invariance of the higher-order differential expression

δ x \cdot [\partial f / \partial x - d (\partial f / \partial d x)]

by means of the identity

^{34}

δ f (x, d x) = δ x \cdot \partial f / \partial x + d (δ x \cdot \partial f / \partial d x) - δ x \cdot d (\partial f / \partial d x)

The conditions (41) directly exploit this invariance for

f (x, d x) = g_{μ ν} (x) d x^{μ} d x^{ν}

; and Riemann may have arrived to the expression

X

through a second-order extension of Lagrange's procedure. The simplest extension would rely on the second-order variation

δ δ (g_{μ ν} d x^{μ} d x^{ν})

. Riemann may have been led to the more complex

X

in an attempt to avoid the third order differentials

δ δ d x^{μ} .35

We will now try to imagine a more geometrical route to the quantity

X

. Riemann surely knew that his symbols (

ι ι^{'}, ι^{''} ι^{'''})

, alias

- 2 R_{μ ν ρ σ}

, were proportional to the coefficients in the form (29) of geodetic distortion he had reached in 1854 in normal geodesic coordinates. This awareness plausibly led him to consider the expression

R_{μ ν ρ σ} (d x^{μ} δ x^{ν} - d x^{ν} δ x^{μ}) (d x^{ρ} δ x^{σ} - d x^{σ} δ x^{ρ})

, which enjoys the following properties: it is of second order with respect to both

d

and

δ

, it is symmetric by permutation of

d

and

δ

, and it vanishes identically when

d x = δ x

. He could then have tried to generate this differential expression by combining double variations of the invariants

g_{μ ν} d x^{μ} d x^{ν}, g_{μ ν} d x^{μ} δ x^{ν}

, and

g_{μ ν} δ x^{μ} δ x^{ν}

. The only combination that shares the properties of the expression is

X

(up to a global coefficient). Riemann could then observe that the third-order differentials canceled out, and could seek to eliminate the second-order differentials in a covariant manner. From a geometric point of view, the obvious means to do so is to draw the geodesics beginning at

x

in the directions of

d x, δ x

, and

d x + δ x

. Riemann's equations (41) immediately result from the vanishing of the variation

δ^{'} \int d s

for these geodesics, and they lead to the expressions (45) for the secondorder differentials at point

x

. Riemann could then verify that under these conditions

X

was proportional to the geodetic distortion of the triangle

x, x + d x, x + δ x

This reasoning raises the question of a direct geometric interpretation of the

X

quantity. The calculation of the distortion of a single geodesic triangle involves differentiations of a different kind (in particular, third-order differentials play a crucial role).

^{36}

We must therefore find something different. I suggest the construction of Figure 2.

From the origin

O

draw the geodesics through two neighboring points

A

and

B

. Mark the points

A^{'}

and

B^{'}

on these geodesics such that

OA = {AA}^{'}

and

OB = {BB}^{'}

. Then draw from

A

and

A^{'}

the two geodesics that make the same angle with the geodesic OA as the geodesic OB does. Similarly, draw from B and

B^{'}

the two geodesics that make the same angle with the geodesic

OB

as the geodesic OA does. The intersection points

O^{'}, A^{''}

, and

B^{''}

define a small geodesic triangle which would be identical with the triangle OAB if the space were flat. In a curved space, the length

A^{''} B^{''}

slightly differs from the length

AB

. I will now show

Figure 2. A geometric interpretation of Riemann's

δ δ (g_{μ ν} d x^{μ} d x^{ν}) - 2 d δ (g_{μ ν} d x^{μ} δ x^{ν}) + dd (g_{μ ν} δ x^{μ} δ x^{ν})

in a non-rigorous manner that the difference of the square of these two lengths is half of Riemann's

X

at the lowest order of approximation.

The triangles being small we have

\begin{aligned} {AB}^{2} \approx {OA}^{2} + {OB}^{2} - 2 \vec{OA} \cdot \vec{OB}, A^{''} B^{'' 2} \approx O^{'} A^{'' 2} + O^{'} B^{'' 2} - 2 \vec{O^{'} A^{''}} \cdot \vec{O^{'} B^{''}} \\ A^{''} B^{'' 2} - {AB}^{2} \approx (O^{'} A^{'' 2} - {OA}^{2}) + (O^{'} B^{'' 2} - {OB}^{2}) - 2 (\vec{O^{'} A^{''}} \cdot \vec{O^{'} B^{''}} - \vec{OA} \cdot \vec{OB}) \end{aligned}

x

denotes the coordinates of

O, x + d x

the coordinates of

A

, and

x + δ x

the coordinates of

B

, we have

{OA}^{2} = {AA}^{' 2} = g_{μ ν} (x) d x^{μ} d x^{ν}, {OB}^{2} = {BB}^{' 2} = g_{μ ν} (x) δ x^{μ} δ x^{ν}, \vec{OA} \cdot \vec{OB} = g_{μ ν} (x) d x^{μ} δ x^{ν}

. The equality of

OA

and

{AA}^{'}

, the equality of

OB

and

{BB}^{'}

, and the equality of the angles marked on the figure correspond to Riemann's conditions (41) or to Eqs. (45). Under these conditions,

O^{'} A^{'' 2} - {OA}^{2}, O^{'} B^{'' 2} - {OB}^{2}

, and

\vec{O^{'} A^{''}} \cdot \vec{O^{'} B^{''}} - \vec{OA} \cdot \vec{OB}

can only be of second order. In addition, the symmetry of the role played by the two geodesics OA and OB implies that

A^{''} B^{'' 2} - {AB}^{2}

should be symmetric by permutation of

d

and

δ

. In sum, we have

A^{''} B^{'' 2} - {AB}^{2} \approx \frac{1}{2} [δ δ (g_{μ ν} d x^{μ} d x^{ν}) + dd (g_{μ ν} δ x^{μ} δ x^{ν}) - 2 d δ (g_{μ ν} d x^{μ} δ x^{ν})] = X / 2

Whether Riemann was aware of this or another geometrical interpretation of his

X

quantity is an open question. Most likely, he followed an intermediate route in which geometrical considerations made him focus on the form

R_{μ ν ρ σ} (d x^{μ} δ x^{ν} - d x^{ν} δ x^{μ}) (d x^{ρ} δ x^{σ} - d x^{σ} δ x^{ρ})

and algebraic considerations of covariance motivated him to obtain this form by varying the invariant fundamental form. Geodetic considerations may also have inspired his way of eliminating the second-order differentials.

2.3. Riemann's private calculations

The file Cod Ms B. Riemann 9 in the Göttingen University Archive contains various materials related to the Commentatio. Most relevant to the present study are a few sheets of calculations in Riemann's hand. I wrote the preceding paragraph before I saw these calculations. I will now examine whether they enable us to narrow down the possible reconstructions of Riemann's train of thoughts.

First consider the two extracts of folio 7 reproduced in Figure 3.

On the left-hand extract, Riemann considers the variable change

(s_{1}, s_{2}, s_{3}) \to (x_{1}, x_{2}, x_{3})

through which

p_{1}^{2} d s_{1}^{2} + p_{2}^{2} d s_{2}^{2} + p_{3}^{2} d s_{3}^{2}

becomes

x_{1}^{2} + x_{2}^{2} + x_{3}^{3} \equiv \sum x^{2}

. He then finds the counterpart of Eqs. (33)-(35) in

Figure 3. Extracts of folio 7 of Cod. Ms. B. Riemann 9; courtesy of Göttingen University Archive.

the special case in which

g_{μ ν}

is diagonal (his

s

variables correspond to my

x

variables). The second extract is the generalization of this result for an arbirary

g_{μ ν}

(Riemann's

a_{ι ι^{'}}

). Remarkably, these calculations begin with a box in which we find the variational equation

δ \int \sqrt{p_{1}^{2} d s_{1}^{2} + p_{2}^{2} d s_{2}^{2} + p_{3}^{2} d s_{3}^{2}} = 0

that gives the geodesics of a diagonal metric. It is not clear whether Riemann used this geometric interpretation to go from Eq. (33) to Eq. (36). It is nonetheless certain that he had this interpretation in mind.

The upper section of folio 35 contains calculations leading from Eq. (36) to the condition

R_{μ v ρ σ} = 0

. These calculations do not differ from those of the final manuscript. There also are several sheets concerning the mysterious

X = δ δ (g_{μ ν} d x^{μ} d x^{ν}) - 2 d δ (g_{μ ν} d x^{μ} δ x^{ν}) + dd (g_{μ ν} δ x^{μ} δ x^{ν})

. The most instructive is folio 9, partly reproduced in Figure 4. Immediately after writing the transformation of the form

\sum α_{ι^{'}} d x_{l} d x_{ι^{'}}

into the form

\sum β_{ι^{'}} d s_{l} d s_{ι^{'}}

, Riemann derives the equation of geodesics of the metric

α_{ι^{'}}

from

δ \int \sqrt{\sum α_{ι^{'}} d x_{l} d x_{l^{'}}} = 0

. He then notes that this equation is directly given by the invariant condition

δ \sum α_{ι ι^{'}} d x_{l} d x_{ι^{'}} - 2 d \sum α_{ι ι^{'}} d x_{l} δ x_{l^{'}} = 0

. The evident purpose of these considerations is to establish the invariance of the conditions

\sum a_{ι^{'}} d^{2} s_{ι^{'}} = \frac{1}{2} \sum p_{ι^{'} ι^{''}} d s_{ι^{'}} d s_{ι^{''}}

, which he is going to use in the foregoing calculation of

X

(built from the

β_{u^{'}}

form, alias

a_{u^{'}}

The expression of the

X

quantity follows the expression

(\partial^{2} β_{l^{'} ι^{'}} / \partial s_{l} \partial s_{l} - 2 \partial^{2} β_{ι^{'}} / \partial s_{l} \partial s_{l^{'}} +

\partial^{2} β_{u l} / \partial s_{l^{'}} \partial s_{l^{'}}) {(d s_{l} δ s_{l^{'}} - d s_{ι^{'}} δ s_{l})}^{2}

, which Riemann most likely obtained by considering the diagonal contribution

(ι ι^{'}, ι^{'}) {(d s_{ι} δ s_{ι^{'}} - d s_{ι^{'}} δ s_{ι})}^{2}

to the geodetic distortion

(ι ι^{'}, ι^{''} ι^{'''}) (d s_{ι} δ s_{ι^{'}} - d s_{ι^{'}} δ s_{ι}) (d s_{ι^{''}} δ s_{ι^{'''}} - d s_{ι^{'''}} δ s_{ι^{''}})

as he knew it from his habilitation lecture (the terms containing the first order derivatives of the metric vanish in the geodesic normal coordinates used in this lecture). He then tried to obtain an invariant generalization of this expression by combining double variations of the metric. The only such combination that vanishes for

d s_{ι} = δ x_{ι}

δ δ \sum β_{ι^{'}} d x_{ι} d x_{ι^{'}} - 2 d δ \sum β_{ι^{'}} d x_{ι} δ x_{ι^{'}} + dd \sum β_{ι^{'}} δ x_{ι} δ x_{ι^{'}}

. Riemann then performed the variations under the invariant conditions (45).

Folio 35, partly reproduced in Figure 5 and probably written at a later time because its contents are closer to the final manuscript, confirms Riemann's use of geodesic considerations in arriving at the invariant conditions (41) and (45) for computing

X

, as well as his focus on the second-order terms of

X

in connecting it with geodetic distortion. Indeed Riemann first gives the geodetic derivation of Eqs. (46). Then

\begin{aligned} \sum α_{c, c} δ_{x_{i}} δ_{x_{c}} = \sum β_{L_{1}}, d_{j_{c}} δ_{j_{i}} \\ \sum_{i} α_{i,}^{'} ϕ_{i}^{''} + \sum \frac{\partial α_{i,}^{'}}{\partial x_{i}^{''}} x_{i}^{'} x_{i}^{'} = \frac{1}{2} Σ^{'} \frac{\partial α_{i,} c^{''}}{\partial x_{i}^{'}} x_{i}^{'} x_{c^{''}}^{'} \end{aligned}

\begin{aligned} (\frac{\partial r_{c_{i}}}{\partial r_{i} \partial r_{i}} - \frac{2 \partial β_{i_{i}}}{\partial r_{i} \partial r_{i}} + \frac{\partial d_{i} i_{i}}{\partial r_{c} \partial_{i}}) {(d_{i} δ_{i} r_{r_{1}} - d_{r_{i}} δ_{i})}^{2} \\ \frac{t_{i}}{} δ \int \sum_{L, i} y d_{i} d_{i} 1 = \sum \\ - 2 d δ \sum ρ_{c_{1}} d_{i} d_{i} δ_{ζ_{l}} \\ + d d \sum_{i_{i} i} δ_{s_{l}} δ_{s_{l}} \end{aligned}

\begin{aligned} + \sum_{a_{i}} (2 \iint_{i}^{2} d_{i} d_{i} + 2 δ^{2} d_{i} d_{i} + 4 δ d s_{i} δ d i_{i} x^{- 2} \end{aligned}

\begin{aligned} d^{t} {\bar{I}}_{a_{4}} d^{c} d i d_{i} ε_{i} = \end{aligned}

Figure 4. Extract of folio 9 of Cod. Ms. B. Riemann 9; courtesy of Göttingen University Archive.

Figure 5. Extract of folio 35 of Cod. Ms. B. Riemann 9; courtesy of Göttingen University Archive.

he gives the terms involving the second order derivatives of

a_{ι^{'}}

for the three double variations of which

X

is the sum. Lastly, he computes

X

by directly exploiting the invariant conditions (41) and bringing out the antisymmetric combinations

d s_{l} δ s_{l^{'}} - d s_{l^{'}} δ s_{l}

as soon as possible.

^{37}

A last interesting sheet is folio 4 (see Figure 6), which contains the Ricci-Bianchi identity

^{38} D_{τ} R_{μ ν ρ σ} +

D_{ρ} R_{μ ν σ τ} + D_{σ} R_{μ ν τ ρ} = 0

. Riemann writes:

\frac{ι^{'}, ι^{''} ι^{'''}}{ι^{I V}} + \frac{ι^{'} ι^{''} ι^{'''} ι^{I V}}{ι} + \frac{ι^{'} ι^{I V} ι^{'''} ι}{ι^{''}} = \sum p_{v, ι^{''} ι^{'''}} (v^{'} ι^{'} ι^{I V}) + p_{v ν ι^{'}} (v^{'} ι^{''} ι^{'''} ι^{I V}) + p_{v ι^{'} ι^{''}} \dots,

which is a schematic rendering of the truncated development

\partial_{τ} R_{μ ν ρ σ} + \partial_{ρ} R_{μ ν σ τ} + \partial_{σ} R_{μ ν τ ρ} = Γ_{τ μ}^{λ} R_{λ ν ρ σ} + Γ_{τ ν}^{λ} R_{μ λ ρ σ} + \dots

Figure 6. Extract of folio 4 of Cod. Ms. B. Riemann 9; courtesy of Göttingen University Archive.

of the Bianchi identity. This interpretation is obtained by decoding the other equations of the extract:

(23) = (1112) \propto R_{1112}

(this is the Ricci convention for the curvature components in a space of three dimensions),

(ι^{'} ι^{''} ι^{'''}) \propto R_{ι ι^{'''} ι^{'} ι^{''}}

(this parenthesis differs from the one used in the Commentatio by a permutation of the indices),

\frac{ι^{'}}{ι^{''} ι^{'''}} = \frac{\partial^{2} a_{ι^{'}}}{\partial s_{ι^{''}} s_{ι^{''}}}

. Clearly, Riemann first obtained

\partial_{τ} R_{μ ν ρ σ}^{(2)} + \partial_{ρ} R_{μ ν σ τ}^{(2)} + \partial_{σ} R_{μ ν τ ρ}^{(2)} = 0

for the second-order contribution

R_{μ ν ρ σ}^{(2)}

to the metric tensor, and then computed the other contributions to

\partial_{τ} R_{μ ν ρ σ} + \partial_{ρ} R_{μ ν σ τ} + \partial_{σ} R_{μ ν τ ρ}

From this examination of Riemann's private calculations, we may conclude that he followed the intermediate route that I judged most probable before seeing them: he combined algebraic manipulations with a focus on geometrically meaningful quantities. We also see that he privately anticipated some features of tensor calculus such as covariant coordinates (see note 37) and the Bianchi identity. He does not seem to have contemplated a geometric interpretation of the mysterious

X

, and he did not do the sort of calculations that Dedekind imagined in the reconstruction that is the object of my next section.

3. After Riemann

3.1. Dedekind

The earliest reader of Riemann's unpublished writings on geometry and quadratic differential forms was his friend Richard Dedekind, who was in charge of his Nachlass after his early death in 1866. Dedekind's reaction can be judged from manuscript notes entitled "Analytical investigations about Riemann's treatises on the hypotheses which lie at the basis of geometry" and probably written while he was preparing the publication of the habilitation lecture.

^{39}

As suggested by the plural in "treatises," part of Dedekind's notes concern the Commentatio, of which he could find drafts in Riemann's papers.

^{40}

The Dedekind manuscript now conserved in the Göttingen University Archive contains an outline and the development of a few points of this outline (Dedekind, 1990). The remaining points are developed in the text published in 1876 by the chief editor of Riemann's collected works, Heinrich Weber, as a commentary to the Commenta- tio (Dedekind and Weber, 1876). The part of Dedekind's manuscript on which Weber says he based his commentary is now lost.

In the extant part, Dedekind introduces invariant measures such as

\sqrt{g_{μ ν} d x^{μ} d x^{ν}}

and

\sqrt{\det g} d^{n} x

(1990,

§ § 2 - 3

); he explains Riemann's concept of immersed plane directions

d x^{μ} δ x^{ν} - δ x^{μ} d x^{ν}

(

§ $ 4 - 5

); he discusses immersed lines, geodesics, and the more general notion of submanifold of extremal content (

$ 8

); and he gives a covariant generalization of Green's theorem (

§ § 6 - 7)

. The latter result, one of his most impressive, reads

\int_{Ω} g^{μ ν} \partial_{μ} u \partial_{ν} v \sqrt{\det g} d^{n} x = \int_{\partial Ω} v \frac{d u}{d N} \sqrt{\det g} d^{n - 1} x - \int_{Ω} v Δ u \sqrt{\det g} d^{n} x

in a modernized notation in which

u

and

v

are two differentiable scalar fields,

Ω

is a bounded domain of the manifold,

\partial Ω

its boundary,

d / d N

the normal derivative at a point of this boundary,

\sqrt{\det g} d^{n - 1} x

the invariant hypersurface element at this point, and

Δ u

is the generalized Laplacian defined by

Δ u = \frac{1}{\sqrt{\det g}} \partial_{μ} (g^{μ ν} \partial_{ν} u \sqrt{\det g})

Dedekind obtained this result by means of the identity

g^{μ ν} \partial_{μ} u \partial_{ν} v \sqrt{\det g} = \partial_{μ} (g^{μ v} v \partial_{ν} u \sqrt{\det g}) - v Δ u \sqrt{\det g}

and of the generalization

\int_{Ω} \partial_{μ} w^{μ} d^{n} x = \int_{\partial Ω} w_{N} d^{n - 1} x

of the divergence theorem. He did not fail to notice that his theorem implied the invariance of the generalized Laplacian.

^{41}

He was probably aware of the connection of this problem with another treated at the beginning of Riemann's Commentatio: the effect of a change of coordinates on the equation of propagation of heat.

The main purpose of Dedekind's commentary to the Commentatio, as we may infer from Weber's rendering, was to justify Riemann's covariant recipe (74) for constructing the curvature system

R_{μ v ρ σ}

. Weber first proves that for the normal coordinates of Riemann's lecture the quantities

Γ_{ν ρ}^{μ}

and

\partial_{σ} Γ_{ν ρ}^{μ} + \partial_{ν} Γ_{ρ σ}^{μ} + \partial_{ρ} Γ_{σ v}^{μ}

vanish at the origin, so that Riemann's conditions (41) for computing

X

reduce to the vanishing of the second-order differentials

dd x, δ δ x

, and

d δ x

, and the three terms of

X

become identical (Dedekind and Weber, 1876, 384-386):

δ δ (g_{μ ν} d x^{μ} d x^{ν}) = - 2 d δ (g_{μ ν} d x^{μ} δ x^{ν}) = dd (g_{μ ν} δ x^{μ} δ x^{ν}) .

As the vanishing of the

Γ

coefficients also implies the vanishing of the first derivatives of

g_{μ ν}

, the geodetic distortion in normal coordinates reads

δ d s^{2} = [g_{μ ν} (δ x) - g_{μ ν} (0)] d x^{μ} d x^{ν} \approx \frac{1}{2} \partial_{ρ} \partial_{σ} g_{μ ν} d x^{μ} d x^{ν} δ x^{ρ} δ x^{σ} = \frac{1}{2} δ δ (g_{μ ν} d x^{μ} d x^{ν}) .

Combining Eqs. (57)-(58), Weber finds (p. 387)

^{42}

δ d s^{2} = X / 6 = \frac{2}{3} \partial_{ρ} \partial_{σ} g_{μ ν} (d x^{μ} δ x^{ρ} - δ x^{μ} d x^{ρ}) (d x^{ν} δ x^{σ} - δ x^{ν} d x^{σ})

in conformity with Eq. (27). He also verifies (pp. 389-391) that in the same coordinates Riemann's

Y =

- (X / 2) {[g_{μ ν} d x^{μ} d x^{ν} (g_{ρ σ} δ x^{ρ} δ x^{σ}) - {(g_{μ ν} d x^{μ} δ x^{ν})}^{2}]}^{- 1}

is the Gaussian curvature of the surface defined by the geodesics tangent to the plane defined by the vector

d x

and

δ x

, in conformity with Riemann's interpretation of Eq. (43). Owing to the invariance of

X

, this interpretation of

Y

remains valid in any coordinate system.

As for the explicit calculation of

R_{μ ν ρ σ}

under Riemann's prescription, Weber only indicates (p. 388) that Riemann's conditions (41) implies the expression (45) of the second-order differentials, "from which one easily gets the expression"43 (42) of the

X

quantity. Weber confused the reader by presenting Riemann's prescription as a covariant generalization of the calculation of geodetic distortion, replacing the vanishing of the second order differentials with the conditions (41) and replacing the noncovariant vanishing of

Γ_{ν ρ}^{μ}

and

\partial_{σ} Γ_{ν ρ}^{μ} + \partial_{ν} Γ_{ρ σ}^{μ} + \partial_{ρ} Γ_{σ v}^{μ}

with the covariant condition

δ δ (g_{μ ν} d x^{μ} d x^{ν}) =

- 2 d δ (g_{μ ν} d x^{μ} δ x^{ν}) = dd (g_{μ ν} δ x^{μ} δ x^{ν})

. This does not make much sense, because this condition and the interpretation of

δ δ (g_{μ ν} d x^{μ} d x^{ν})

as twice the geodetic distortion dot not hold in arbitrary systems of coordinates.

An early reader of this first edition of Riemann's Commentatio, Richard Beez, complained that "Riemann's prescription did not get the desired result" 44 (Beez, 1879, 9-10). Beez compared the second-order terms of Riemann's

X

, which are

X^{(2)} = (\partial_{ρ} \partial_{σ} g_{μ ν} + \partial_{μ} \partial_{ν} g_{ρ σ} - 2 \partial_{ν} \partial_{σ} g_{μ ρ}) d x^{μ} d x^{ν} δ x^{ρ} δ x^{σ},

with the second-order terms of the Riemann tensor, which are

- 2 R_{μ ρ v σ}^{(2)} = \partial_{ρ} \partial_{σ} g_{μ ν} + \partial_{μ} \partial_{ν} g_{ρ σ} - \partial_{μ} \partial_{σ} g_{ρ v} - \partial_{ρ} \partial_{ν} g_{μ σ}

and he declared them to be incompatible. In reality, the two expressions are perfectly compatible, because Riemann requires only the relation (42),

X = - \frac{1}{2} R_{μ ρ v σ} (d x^{μ} δ x^{ρ} - d x^{ρ} δ x^{μ}) (d x^{ν} δ x^{σ} - d x^{σ} δ x^{ν})

whose second-order part results from Eqs. (60)-(61) because

\partial_{μ} \partial_{σ} g_{ρ ν}

and

\partial_{ρ} \partial_{ν} g_{μ σ}

are equivalent to

\partial_{ν} \partial_{σ} g_{μ ρ}

when contracted with

d x^{μ} d x^{ν} δ x^{ρ} δ x^{σ}

and because the symmetries of

R_{μ ρ v σ}

imply that

R_{μ ρ ν σ} d x^{μ} d x^{ν} δ x^{ρ} δ x^{σ}

equals

4 R_{μ ρ ν σ} (d x^{μ} δ x^{ρ} - d x^{ρ} δ x^{μ}) (d x^{ν} δ x^{σ} - d x^{σ} δ x^{ν})

. In the second edition of Riemann's collected paper, published in 1892, Weber attributed Beez's and others' confusion to the excessive concision of his and Dedekind's commentary, and he produced a more detailed version in which the prescription for computing

X

was made clearer (Weber, 1892, 410-411).

On the one hand, Dedekind and Weber's commentary was mostly analytical, in conformity with the title of Dedekind's manuscript: the main purpose was to explicate the calculations that Riemann did not find

terms that do not contribute to

X

43 "woraus man leicht den Ausdruck erhält [...]".

44 "Man überzeugt sich nun leicht, dass man bei stricter Befolgung der Riemman'schen Vorschrift nicht zu dem gewünschten Ziele gelangt."
}
time to write down or publish. On the other hand, this commentary emphasized the geometrical aspects of the second part of the Commentatio by relating it both to Gaussian curvature (as Riemann succinctly did) and to geodetic distortion (as Riemann did not). They thus inaugurated a tradition of regarding the Commentatio as an opportunity for Riemann to summarize some of the calculations he must have done to prepare his habilitation lecture. In his introduction, Weber (1876, IV) announced this text as being "of especially high interest because therein Riemann's investigations about the general properties of multiply extended manifolds are recorded in their main features and find a remarkable application." 45

3.2. Christoffel

In 1869, two years after the publication of Riemann's habilitation lecture, Dedekind's successor at the Zürich Polytechnicum, Elwin Bruno Christoffel, published a powerful memoir on the mutual transformability of two quadratic differential forms. As Christoffel explained in his introduction, "[his] researches originated in the

n

-dimensional extension of the problem of surfaces that can be developed onto one another" 46 (Christoffel, 1869, 47). The latter problem had motivated Gauss's Disquisitiones, and its

n

-dimensional generalization could easily be conceived through Riemann's concept of a metric manifold. At the end of his memoir (p. 69), Christoffel mentioned the recent publication of Riemann's lecture of 1854 as well as the existence of a manuscript in which Dedekind gave the relevant calculations.

^{47}

If Christoffel saw this manuscript, then he was aware of Riemann's handling of the transformability problem in the Commentatio. If he did not, he was at least aware of the geometrical handling of a particular case of this problem in the lecture of 1854. Yet his subsequent considerations were formulated in a purely algebraic manner.

^{48}

Christoffel's problem is to find under what conditions the form

g_{μ ν} (x) d x^{μ} d x^{ν}

can be obtained from a given form

g_{μ ν}^{'} (x^{'}) d x^{' μ} d x^{' ν}

through a variable change

x^{'} \to x

. In symbols the effect of a variable change is

g_{μ ν} [x (x^{'})] \frac{\partial x^{μ}}{\partial x^{'} μ^{'}} \frac{\partial x^{ν}}{\partial x^{' v^{'}}} = g_{μ ν}^{'} (x^{'}) .

Being an expert on the algebraic theory of invariants, Christoffel tried to replace this partial differential equation with purely algebraic conditions for the covariance of certain multilinear forms. A first algebraic condition is the existence, for any given value of

x^{'}

, of

n

numbers

x^{μ}

and

n^{2}

numbers

u_{μ^{'}}^{μ}

such that

g_{μ v} (x) u_{μ^{'}}^{μ} u_{v^{'}}^{v} = g_{μ^{'} v^{'}}^{'} (x^{'}) .

If there were only one choice of these numbers, say

x^{μ} (x^{'})

and

u_{μ^{'}}^{μ} (x^{'})

, for which the condition is satisfied and if it could be shown that

u_{μ^{'}}^{μ} = \partial x^{μ} / \partial x^{' μ^{'}}

, then the problem would be solved. This is unfortunately not the case, because any quadratic form can be turned into any other quadratic form with the same signature by linear transformations of the kind (63). In order to remedy this indeterminacy, Christoffel constructs other algebraic conditions of the form

T_{μ ν ρ σ \dots} (x) u_{μ^{'}}^{μ} u_{ν^{'}}^{v} u_{ρ^{'}}^{ρ} u_{σ^{'}}^{σ} \dots = T_{μ^{'} ν^{'} ρ^{'} σ^{'} \dots}^{'} (x^{'})

through successive differentiations of Eq. (62). In modern terms, he derives tensors of higher order by repeated differentiation of the metric tensor.

^{49}

By a first differentiation of Eq. (62) and by combining index-permuted versions of the result, Christoffel obtains

(1869, 49)

\frac{\partial^{2} x^{μ}}{\partial x^{'} v^{'} \partial x^{'} ρ^{'}} + Γ_{v ρ}^{μ} (x) \frac{\partial x^{v}}{\partial x^{' v^{'}}} \frac{\partial x^{ρ}}{\partial x^{'} ρ^{'}} = Γ_{v^{'} ρ^{'}}^{μ^{'}} (x^{'}) \frac{\partial x^{μ}}{\partial x^{'} μ^{'}}

with the definition (11) of the

Γ

coefficients.

^{50}

A second differentiation and the permutability of two successive partial derivatives yield the compatibility condition

\frac{\partial}{\partial x^{' σ^{'}}} [Γ_{v ρ}^{μ} (x) \frac{\partial x^{ν}}{\partial x^{' v^{'}}} \frac{\partial x^{ρ}}{\partial x^{'} ρ^{'}} - Γ_{v^{'} ρ^{'}}^{μ^{'}} (x^{'}) \frac{\partial x^{μ}}{\partial x^{'} μ^{'}}] = \frac{\partial}{\partial x^{'} ρ^{'}} [Γ_{v ρ}^{μ} (x) \frac{\partial x^{ν}}{\partial x^{'} v^{'}} \frac{\partial x^{ρ}}{\partial x^{' σ^{'}}} - Γ_{v^{'} σ^{'}}^{μ^{'}} (x^{'}) \frac{\partial x^{μ}}{\partial x^{'} μ^{'}}],

from which Christoffel laboriously derives

(1869, 54)

R_{μ ν ρ σ} (x) u_{μ^{'}}^{μ} u_{ν^{'}}^{v} u_{ρ^{'}}^{ρ} u_{σ^{'}}^{σ} = R_{μ^{'} ν^{'} ρ^{'} σ^{'}}^{'} (x^{'})

with the definition (17) of

R_{μ ν ρ σ}

Christoffel does not relate these coefficients to Riemann's curvature, even though he builds the symmetric quadrilinear form

R_{μ ν ρ σ} (d x^{μ} δ x^{ν} - d x^{ν} δ x^{μ}) (D x^{ρ} Δ x^{σ} - D x^{σ} Δ x^{ρ})

, which is the polar of Riemann's form (29) or (42). His notation for

R_{μ v ρ σ}

is (

g k h i)

(i i_{1} i_{2} i_{3}

), not too different from the (

ι ι^{'}, ι^{''} ι^{'''})

of Riemann's Commentatio, although this may be a coincidence.

Christoffel

(1869, § 6)

next shows that for any

T_{μ ν ρ σ \dots}

and

T_{μ^{'} ν^{'} ρ^{'} σ^{'} \dots}^{'}

such that

T_{μ ν ρ σ \dots} (x) u_{μ^{'}}^{μ} u_{ν^{'}}^{v} u_{ρ^{'}}^{ρ} u_{σ^{'}}^{σ} \dots = T_{μ^{'} ν^{'} ρ^{'} σ^{'} \dots}^{'} (x^{'})

, the form

D_{λ} T_{μ ν ρ σ \dots} = \partial_{λ} T_{μ ν ρ σ \dots} - Γ_{λ μ}^{κ} T_{κ ν ρ σ \dots} - Γ_{λ ν}^{κ} T_{μ κ ρ σ \dots} - Γ_{λ ρ}^{κ} T_{μ ν κ σ \dots} - Γ_{λ σ}^{κ} T_{μ ν σ κ \dots} - \dots

transforms according to the same rule. In modern terms, he thus builds the covariant derivative of a covariant tensor of any order. In addition, he proves that the covariant derivative of

g_{μ ν}

vanishes, a result often called "Ricci's lemma." 51

Whereas we now regard these results as the formal basis of tensor calculus on a Riemannian manifold, for Christoffel they were only a means to transform the initial problem of the equivalence of two quadratic differential forms into the algebraic problem of the equivalence of a sequence of multilinear forms. In most cases (when the invariance group of the quadratic differential form is trivial), he expected that the algebraic conditions

g_{μ ν} (x) u_{μ^{'}}^{μ} u_{v^{'}}^{v} = g_{μ^{'} ν^{'}}^{'} (x^{'})

and

T_{μ ν ρ σ \dots} (x) u_{μ^{'}}^{μ} u_{ν^{'}}^{v} u_{ρ^{'}}^{ρ} u_{σ^{'}}^{σ} \dots = T_{μ^{'} ν^{'} ρ^{'} σ^{'} \dots}^{'} (x^{'})

for

T_{μ ν ρ σ \dots} = R_{μ ν ρ σ}, D_{τ} R_{μ ν ρ σ}, D_{v} D_{τ} R_{μ v ρ σ}

, etc. completely determined the values of the numbers

x^{μ}

and

u_{μ^{'}}^{μ}

for any given

x^{'}

if the number of repetitions of the covariant derivation was large enough. Suppose for instance that the conditions involving

g_{μ ν}

and

R_{μ ν ρ σ}

are sufficient to determine these numbers (this is true if

n = 3

and if the group of automorphisms is trivial). Call

x^{μ} (x^{'})

and

u_{μ^{'}}^{μ} (x^{'})

the result of this determination. Then, if the condition involving

D_{τ} R_{μ ν ρ σ}

is also satisfied, the equations

u_{μ^{'}}^{μ} (x^{'}) = \partial x^{μ} / \partial x^{'} μ^{'}

hold automatically. More generally, if a series of conditions of the form

T_{μ \dots} (x) u_{μ^{'}}^{μ} \dots = T_{μ^{'} \dots}^{'} (x^{'})

suffice to determine

x^{μ} (x^{'})

and

u_{μ^{'}}^{μ} (x^{'})

and if, in addition, the derived conditions

D_{λ} T_{μ \dots} u_{λ^{'}}^{λ} u_{μ^{'}}^{μ} \dots = D_{λ^{'}}^{'} T_{μ^{'} \dots}^{'}

also hold, then the equations

u_{μ^{'}}^{μ} (x^{'}) = \partial x^{μ} / \partial x^{'} μ^{'}

hold automatically and the two forms

g_{μ ν}

and

g_{μ ν}^{'}

are equivalent. Christoffel's proof of this remarkable theorem goes as follows

(1869, § § 9 - 10)

Injecting the solutions

x^{μ} (x^{'})

and

u_{μ^{'}}^{μ} (x^{'})

in the determining conditions

T_{μ ν \dots} (x) u_{μ^{'}}^{μ} u_{v^{'}}^{v} \dots = T_{μ^{'} v^{'} \dots}^{'} (x^{'})

and differentiating with respect to

x^{' λ^{'}}

yields

(\partial_{λ} T_{μ ν \dots} u_{μ^{'}}^{μ} u_{v^{'}}^{ν} \dots) (\partial x^{λ} / \partial x^{' λ^{'}}) + T_{μ ν \dots} (\partial u_{μ^{'}}^{μ} / \partial x^{' λ^{'}}) u_{ν^{'}}^{ν} \dots + \dots = \partial T_{μ^{'} ν^{'} \dots}^{'} / \partial x^{' λ^{'}}

The system of all such equations, regarded as a linear non-homogenous system for the unknowns

\partial x^{λ} / \partial x^{'} λ^{'}

and

\partial u_{μ^{'}}^{μ} / \partial x^{' λ^{'}}

must completely determine these unknowns since the system of all Eqs. (69) completely determine the unknowns

x

and

u_{μ^{'}}^{μ}

. Consequently, the homogenous linear system of all the equations

(\partial_{λ} T_{μ ν \dots} u_{μ^{'}}^{μ} u_{v^{'}}^{v} \dots) X_{λ^{'}}^{λ} + T_{μ ν \dots} Y_{μ^{'} λ^{'}}^{μ} u_{ν^{'}}^{v} \dots + \dots = 0

must imply

X_{λ^{'}}^{λ} = 0

and

Y_{μ^{'} λ^{'}}^{μ} = 0

. Such a system is obtained by subtracting

D_{λ} T_{μ \dots} u_{λ^{'}}^{λ} u_{μ^{'}}^{μ} \dots = D_{λ^{'}}^{'} T_{μ^{'} \dots}^{'}

from Eq. (70). In this case, we have

X_{λ^{'}}^{λ} = \partial x^{λ} / \partial x^{' λ^{'}} - u_{λ^{'}}^{λ}

. Consequently,

u_{λ^{'}}^{λ}

must be equal to

\partial x^{λ} / \partial x^{' λ^{'}}

, as was to be proved.

Christoffel's initial problem is thus solved through a purely algebraic set of conditions. As Felix Klein (1926-1927, 2: 197) later noted, the benefit is not as large as it would seem, because the algebraic elimination problem remains difficult. Later mathematicians, especially Elie Cartan, gave more elegant and more general solutions to Christoffel's problem.

^{52}

The tensor-calculus byproducts of Christoffel's investigation are nonetheless considerable.

Since these byproducts later received a geometric interpretation, one may wonder whether Christoffel privately relied on geometric intuition in their derivation. Firstly, he may have reached Eq. (65) by requiring that the geodesics of the

g_{μ ν}

metric should correspond to the geodesics of the

g_{μ ν}^{'}

metric. This is quite plausible because the notation he used for the

Γ

coefficients,

{\begin{matrix} ν ρ \\ μ \end{matrix}}

, is the one he had earlier used to denote the coefficients of the equation of geodesics on a surface (Christoffel, 1868, 126). Secondly, in his discovery of Eq. (67),

R_{μ ν ρ σ} (x) u_{μ^{'}}^{μ} u_{v^{'}}^{v} u_{ρ^{'}}^{ρ} u_{σ^{'}}^{σ} = R_{μ^{'} ν^{'} ρ^{'} σ^{'}}^{'} (x^{'})

, Christoffel may have been inspired by his awareness of the quadratic form that Riemann associated with the

R_{μ ν ρ σ}

coefficients in his lecture of 1854. Some knowledge of the contents of the Commentatio, even indirect, could also have helped him. Assuming that Christoffel privately relied on geometric intuitions and derivations, why would he suppress them from the published memoir? Perhaps he wanted to emphasize continuity with his earlier work on the algebraic theory of invariants. Perhaps he was following the growing trend of arithmetizing or algebraizing geometry and analysis. Perhaps, he did not believe that the study of

n

-dimensional manifolds truly belonged to geometry. Indeed his few mentions of geometrical ideas and problems in his memoir concerned ordinary, two-dimensional surfaces.

^{53}

3.3. Lipschitz

In the same year 1869 and in the same journal, the Bonn mathematician Rudolf Lipschitz published a memoir on the condition for a given homogenous differential form

^{54}

to be equivalent to a form with constant coefficients through a change of variables (Lipschitz, 1869). Despite evident similarities, this problem differs from Christoffel's in two manners. Firstly, Lipschitz considers homogenous forms of any degree (including one), whereas Christoffel regards the degree two (quadratic forms) as the only interesting case. Indeed for any higher degree, the first algebraic condition of equivalence (the counterpart of condition (63)) completely determines the functions

x^{μ} (x^{'})

and

u_{μ^{'}}^{μ} (x^{'})

and there is no need for deriving conditions of higher order. Secondly, Lipschitz treats the equivalence of a differential form with a form with constant coefficients, whereas Christoffel's method excludes this case as well as any case for which the differential form admits non-trivial automorphisms.

^{55}

Lipschitz

(1869, 71 - 72)

acknowledged the stimulus provided by the publication of Riemann's habilitation lecture. In the quadratic case, Riemann had indeed given the condition sought by Lipschitz: the curvature of the metric manifold associated with the quadratic field must vanish. As the full expression of Riemann's curvature appeared only in the unpublished Commentatio, Lipschitz was not quite sure that his own equivalence condition agreed with Riemann's.

^{56}

But he surely understood the importance that Gauss and Riemann had given to such a condition and to its geometric interpretation. We will see that analogy with Gauss's intrinsic expression of curvature guided his construction of the general equivalence condition. However, by generalizing to forms of any degree Lipschitz distanced himself from Gaussian and Riemannian geometry; and his heuristic involved an important non-geometrical component: appeal to Lagrange's calculus of variations in the mechanical tradition continued by William Rowan Hamilton and by Carl Jacobi.

^{57}

Whereas Christoffel used ordinary differentiation only (affecting only the coefficients of a differential form), Lipchitz used variations that increased the differential order of differential forms.

^{58}

As was earlier mentioned, in his Mécanique analytique (1811-1815, 1: 307-310) Lagrange showed how to derive a higher-order invariant differential expression from a lower-order one.

^{59}

For the invariant form

f (x, d x)

(with

x = (x_{1}, x_{2}, \dots, x_{n})

, etc.), this procedure leads to the new invariant form

ω (x, d x, δ x) = δ x \cdot [d (\partial f / \partial d x) - \partial f / \partial x]

As a first application of this rule,

Lipschitz (1869, 77)

considered the linear form

f = a_{μ} d x^{μ}

, whose derived invariant form reads

ω = \partial_{μ} a_{ν} (d x^{μ} δ x^{ν} - d x^{ν} δ x^{μ})

This form vanishes if and only if

\partial_{ν} a_{μ} - \partial_{μ} a_{ν} = 0

, which is the condition for the form

f

to be a differential. Evidently, this condition must hold if there exists a system of coordinates for which the coefficients

a_{μ}

are constant. Reciprocally, if this condition holds, the form

f

can be integrated according to

f = d F (x)

; in a new system of coordinates in which

F

is the first coordinate, the differential form has constant coefficients. This solves Lipschitz's problem for forms of degree one.

In the general case, the derived form is

ω = (\frac{\partial^{2} f}{\partial d x^{μ} \partial d x^{ν}} d^{2} x^{ν} + \frac{\partial^{2} f}{\partial d x^{μ} \partial x^{ν}} d x^{ν} - \frac{\partial f}{\partial x^{μ}}) δ x^{μ}

For the quadratic field

f = g_{μ ν} d x^{μ} d x^{ν}

, this gives

ω = g_{ρ σ} (d^{2} x^{ρ} + Γ_{μ ν}^{ρ} d x^{μ} d x^{ν}) δ x^{σ}, with g_{ρ σ} Γ_{μ ν}^{ρ} = \frac{1}{2} (\partial_{μ} g_{σ ν} + \partial_{ν} g_{σ μ} - \partial_{σ} g_{μ ν})

Lipschitz's next step was to eliminate the second order differentials

d^{2} x^{μ}

in the invariance relation

ω = ω^{'}

, by analogy with the formation of the Gaussian curvature in the case of quadratic differential forms. After complicated calculations involving the transformation rule for

d^{2} x^{μ}

, he obtained the invariance of a fourthorder differential form built from the first and second partial derivatives of the form

f

. In the quadratic case the new form reads (Lipschitz, 1869, 84)

Ω = R_{μ ν ρ σ} d x^{μ} δ x^{ν} d^{'} x^{ρ} δ^{'} x^{σ}, with R_{μ ν ρ σ} = g_{μ v} (\partial_{ρ} Γ_{ν σ}^{v} - \partial_{σ} Γ_{ν ρ}^{v} + Γ_{ρ τ}^{v} Γ_{ν σ}^{τ} - Γ_{σ τ}^{v} Γ_{v ρ}^{τ}),

in conformity with the expression (17) of Riemann's curvature. The invariance of this form implies that it should vanish if there exists a system of coordinates for which the coefficients of the form

f

are constants. Lipschitz most impressive achievement was his proof that reciprocally the vanishing of the form

Ω

implies the existence of a system of coordinates for which the coefficients of the form are constants (Lipschitz,

1869, § 6)

. For this purpose, he relied on uniform geodetic motion to construct the geodetic coordinates for which the form

f

becomes uniform.

In accordance with the mechanical nature of the latter reasoning, Lipschitz developed mechanical aspects of the study of homogenous forms in subsequent memoirs (Lipschitz, 1870a, 1870b). The basic idea was to relate curvature to the inertial force experienced by a particle constrained to move on a curved manifold. He developed the notion of geodesic normal coordinates-which he called central coordinatesagain in a mechanical manner associated with geodetic motion. In this context, he was able to prove Riemann's relation between geodetic distortion and curvature: that is to say, he derived the expression

- \frac{1}{3} R_{μ ν ρ σ} d x^{μ} δ x^{ν} d x^{ρ} δ x^{σ}

of the correction to the length of the element

d x

at the point

x + δ x

in normal coordinates centered on

x

(1870b, 23).

Another of Lipschitz' achievements was the manifestly covariant expression

Ω = δ^{'} x \cdot (D Δ - Δ D) d^{'} x

he gave to the form

Ω

in terms of the covariant variations

D

and

Δ

such that

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D a^{μ} = d a^{μ} + Γ_{ν ρ}^{μ} d x^{ν} a^{ρ} and Δ a^{μ} = δ a^{μ} + Γ_{ν ρ}^{μ} δ x^{v} a^{ρ}

(Lipschitz, 1870b, 16), to be compared with Eqs. (47). In 1877, after reading the Commentatio in Riemann's collected works, he split

Ω

into two invariant parts:

\begin{aligned} Ω = \frac{1}{2} \hat{X} + Ξ, with \\ \frac{1}{2} \hat{X} = D (δ^{'} x \cdot Δ d^{'} x) - Δ (δ^{'} x \cdot {Dd}^{'} x) and Ξ = - D δ^{'} x \cdot Δ d^{'} x + Δ δ^{'} x \cdot {Dd}^{'} x \end{aligned}

\overset{―}{60}

The notation and the naming are mine. Lipschitz instead uses the covariant

ψ

vectors such that

ψ_{μ} (d x, a) = g_{μ ν} D a^{ν}

Using the identities

d (a \cdot b) = D (a \cdot b) = (D a) \cdot b + b \cdot (D a), {Dd}^{'} x = D^{'} d x, D δ^{'} x = Δ^{'} d x

, etc., we have

\hat{X} = {dd}^{'} (δ x \cdot δ^{'} x) + δ δ^{'} (d x \cdot d^{'} x) - δ d^{'} (d x \cdot δ^{'} x) - d δ^{'} (δ x \cdot d^{'} x)

For

d = d^{'}

and

δ = δ^{'}

this expression becomes identical to Riemann's

X

(Lipschitz, 1877, 317-321). As for the invariant

Ξ

, Lipschitz noted that it vanished under Riemann's conditions (41), which are equivalent to

D δ x = 0, Dd x = 0

, and

Δ δ x = 0

. Furthermore, he noted that an invariant of the same type as this one occurred in rational mechanics when Gauss's principle of least constraint was expressed in arbitrary coordinates (Lipschitz, 1877, §§2-3).

In terms of the positions

r_{a}

, masses

m_{α}

, and forces

f_{α}

for the particles

α

of a connected system, this principle reads

δ \sum_{α} m_{α} {({\ddot{r}}_{α} - f_{α} / m_{α})}^{2} = 0

If the positions

r_{a}

depend only on the independent configuration variables

q^{1}, \dots, q^{n}

, the kinetic energy of the system is the quadratic form

T = (1 / 2) g_{i j} {\dot{q}}^{i} {\dot{q}}^{k}

, and the work of the forces during an infinitesimal change of configuration is

\sum_{α} f_{α} \cdot d r_{α} = F_{i} d q^{i}

, which defines the external generalized forces

F_{i}

. Introducing the covariant differentiation D with respect to the quadratic differential form

g_{i j} d q^{i} d q^{j}

, the Lagrangian equation of motion reads

F_{i} = g_{i j} D {\dot{q}}^{j} / d t

, and the principle of least constraints reads

δ [g^{i j} (g_{i k} \frac{D {\dot{q}}^{k}}{d t} - F_{i}) (g_{j k} \frac{D {\dot{q}}^{k}}{d t} - F_{j})] = 0 .

Lipschitz compared the invariant

g_{i j} Dd q^{i} Dd q^{j}

occurring in this expression with the invariants

g_{μ ν} Δ δ x^{μ}

Dd x^{ν}

occurring in the expression of the invariant

Ξ

. With this analogy, the decomposition of the invariant

Ω

became more natural. Lipschitz thus solved the mystery of Riemann's

X

in his own manner, that is, through a theory of differential invariants based on variational calculus in mechanical context.

^{61}

3.4. Ricci

Lipschitz was not alone in connecting the theory of quadratic differential forms to analytical mechanics. So did too Eugenio Beltrami in 1868, Ernst Christian Julius Schering in 1873, Wilhelm Killing in 1885, and Heinrich Hertz in

1894 .^{62}

For others including Dedekind, Aurel Voss, Richard Beez, Felix Klein, Hermann Helmholtz, and Sophus Lie, the geometrical interpretation was more important. A third tendency, inaugurated by Christoffel, was to develop the theory of differential invariants for its own sake, independently of any application. The Padova mathematician Gregorio Ricci-Curbastro was aware of these three approaches when he began working on the theory of quadratic differential forms in the mid-1880s. Although he was himself interested in geometrical, mechanical, and physical applications of this theory, he believed that for the sake of rigor and generality foundations should precede applications

^{63}

(Ricci-Curbastro, 1884, 140):

The aim of this writing is to initiate a series of researches on the theory of quadratic differential forms; being founded on purely analytical concepts, these researches will best introduce us into the knowledge of the nature of these forms and will thus avoid all idle discussions about the existence and nature of spaces of more than three dimensions. Interpretations dictated by geometrical or mechanical analogies-such interpretations can indeed be given — will only be illustrations of such a theory.

^{64}

In his first memoir on this topic, published in Italian in 1884, Ricci defined the class of a quadratic differential form of

n

variables as the number

h

such that

n + h

is the minimal number of variables of a form with constant coefficients from which the original form can be derived by constraining these variables through

h

relations. Ricci indeed knew from Ludwig Schlaefli that there always is an Euclidean space of dimension

n + h

of which a given metric manifold of dimension

n

is a submanifold, with

0 ⩽ h ⩽ n (n - 1) / 2

. Ricci then proved that the vanishing of Riemann's

R_{μ ν ρ σ}

was a necessary and sufficient condition for a form to be of class zero, and he found similar conditions for forms of class one (Ricci-Curbastro, 1884). In a series of subsequent memoirs (Ricci-Curbastro, 1886, 1887, 1888, 1892), he thoroughly developed an "absolute differential calculus" (calcolo differenziale assoluto) containing all the basic definitions and operations of modern tensor analysis on differential manifolds: covariant and contravariant "systems of functions" (sistemi di funzioni, our tensors), tensor products, contraction, raising and lowering of indices through the fundamental (metric) tensor, covariant and contravariant derivation, the Riemann curvature tensor and its relation to the commutator of two covariant derivations. He also offered a number of geometrical and physical applications of his new calculus, including systems of geodesics, congruences, elasticity, heat propagation in curved hyperspaces, and continuous groups of motion. The most systematic exposition of his theory was the one he published in French in the Mathematische Annalen in 1901 with his brilliant disciple Tullio Levi-Civita (Ricci-Curbastro and Levi-Civita, 1901).

3.5. Levi-Civita, parallel transport, and Riemann

In the 1890's, young Levi-Civita had abundantly used Ricci's calculus in contributions to the theory of differential invariants, to its application to mechanics, and to its connection with Lie's theory of continuous groups. He returned to this topic in 1917, after Albert Einstein's publication of the general theory of relativity. Einstein's impressive application of Riemann's concept of metric manifold and of Ricci's absolute differential calculus induced Levi-Civita to "somewhat reduce the formal apparatus that usually serves to introduce and establish the covariant behavior" 65 (Levi-Civita, 1917, 173). For this purpose, he sought inspiration in Riemann's writings.

^{66}

As we saw, the condition (41) of Riemann's Commentatio includes the covariant relation

δ d x^{μ} = d δ x^{μ} = - Γ_{v ρ}^{μ} δ x^{ν} d x^{ρ}

between the second-order differential

δ d x^{μ}

and the first order differentials

δ x^{μ}

and

d x^{μ}

. After emphasizing the covariant character of this relation, Levi-Civita considered the third order differential

\begin{aligned} δ^{'} δ d x^{μ} & = - δ^{'} (Γ_{v ρ}^{μ} δ x^{ν} d x^{ρ}) = - δ^{'} Γ_{ν ρ}^{μ} δ x^{ν} d x^{ρ} - 2 Γ_{v ρ}^{μ} δ^{'} δ x^{ν} d x^{ρ} \\ = - \partial_{σ} Γ_{v ρ}^{μ} δ x^{ν} d x^{ρ} δ^{'} x^{σ} + 2 Γ_{ν ρ}^{μ} Γ_{σ τ}^{ν} d x^{ρ} δ x^{σ} δ^{'} x^{τ} \end{aligned}

the similar differential

δ δ^{'} d x^{μ}

, and he formed the difference

u^{μ} = (δ^{'} δ - δ δ^{'}) d x^{μ}

which he found to be a contravariant vector. Indeed for any covariant vector

p_{μ}

, the combination

p_{μ} u^{μ}

is an invariant scalar owing to the identities

\begin{aligned} δ^{'} δ p_{μ} = δ^{'} (\partial_{ν} p_{μ} δ x^{ν}) & = \partial_{ν} \partial_{ρ} p_{μ} δ x^{ν} δ^{'} x^{ρ} + \partial_{ν} p_{μ} δ^{'} δ x^{ν} = δ δ^{'} p_{μ} \\ (δ^{'} δ - δ δ^{'}) (p_{μ} d x^{μ}) & = δ^{'} (p_{μ} δ d x^{μ} + δ p_{μ} d x^{μ}) - δ (p_{μ} δ^{'} d x^{μ} + δ^{'} p_{μ} d x^{μ}) \\ = p_{μ} u^{μ} + d x^{μ} (δ^{'} δ - δ δ^{'}) p_{μ} = p_{μ} u^{μ} \end{aligned}

The effective calculation of

u^{μ}

yields

u^{μ} = R^{μ}_{ν ρ σ} d x^{ν} δ x^{ρ} δ^{'} x^{σ} .

Consequently, the expression

I = g_{μ τ} d^{'} x^{τ} (δ^{'} δ - δ δ^{'}) d x^{μ} = R_{μ ν ρ σ} d^{'} x^{μ} d x^{ν} δ x^{ρ} δ^{'} x^{σ} with R_{μ ν ρ σ} = g_{μ τ} R_{ν ρ σ}^{τ}

is an invariant, and

R_{μ ν ρ σ}

must be a completely contravariant 4-tensor. To which Levi-Civita commented: "If I am not mistaken, this is the fastest way to arrive at this result"

^{67}

(Levi-Civita, 1917, 198).

This simple derivation of the Riemann tensor occurs toward the end of Levi-Civita's memoir, in

§ § 15 - 16

, after an extensive development of the notion of parallel transport. This differs from the order of discovery, which Levi-Civita reveals at the beginning of his memoir: he first obtained the new derivation of the Riemann tensor, and then sought a geometric interpretation of this derivation (Levi-Civita, 1917, 173):

Although this work was originally done with this sole motivation [the simplification of the derivation of covariant behavior found in

§ § 15 - 16]

, in its subsequent expansion it ended up giving a proper place to the geometrical interpretation.

^{68}

As Levi-Civita observes, for

δ = d

the relation (83) has a simple geometric interpretation through the equation of geodesics: for the geodesic passing through

x

and

x + d x, d x

is a tangent vector at

x

, and

x + dd x^{μ} = x - Γ_{ν ρ}^{μ} d x^{ν} d x^{ρ}

is the tangent vector at

x + d x

that has the same length as the vector

d x

. In the general case

δ \neq d

, Levi-Civita (1917, 196, Eqs. 31) was aware of the identities

d (g_{μ ν} δ x^{μ} δ x^{ν}) = 0 and d (g_{μ ν} d x^{μ} δ x^{ν}) = 0

whose geometric interpretation is the conservation of the length of the vector

δ x

and of its angle with the vector

d x

when the vector

δ x

changes by

- Γ_{ν ρ}^{μ} δ x^{ν} d x^{ρ}

along the geodesic line passing through

x

and

x +

d x

. Intuitively, geodesics are the straightest possible lines on the manifold; therefore, on a two-dimensional manifold for which the direction of a vector is completely determined by its angle with a given line, the direction of the vector

δ x + d (δ x)

at point

x + d x

may be said to differ as little as possible from the direction of vector

δ x

at point

x

Figure 7. Levi-Civita's quasi-parallelogram.

Possibly guided by this intuition, Levi-Civita looked for a notion of parallel displacement on the manifold. He found one by regarding the manifold as an hypersurface in an Euclidean space of higher dimension (Levi-Civita, 1917, 174-175, §§2-3). This immersion provides an extrinsic notion of parallelism for two vectors situated at two different points of the manifold: they have to make the same angle with any given vector of the Euclidean space. For intrinsic parallelism, Levi-Civita tried the next best thing: two vectors a and

b

situated at two neighboring points

x

and

x + d x

of the manifold are said to be parallel if and only if they make the same angle with any vector belonging to the tangent hyperplane at point

x

. If

r (x)

denotes the immersion of the point of coordinates

x^{μ}

in the larger Euclidean space, this condition gives

a \cdot \partial_{μ} r = b \cdot \partial_{μ} r to first order in d x for μ = 1, \dots, n

Remembering that

a = a^{μ} \partial_{μ} r (x), b = b^{μ} \partial_{μ} r (x + d x)

, and

g_{μ ν} = \partial_{μ} r \cdot \partial_{ν} r

, this is equivalent to

a^{μ} g_{μ ν} = b^{μ} g_{μ ν} + b^{μ} (\partial_{μ} \partial_{ρ} r) \cdot \partial_{ν} r d x^{ρ} .

Using

(\partial_{μ} \partial_{ρ} r) \cdot \partial_{ν} r = g_{ν σ} Γ_{μ ρ}^{σ}

(Eq. (36)), we get

a^{μ} = b^{μ} + Γ_{ν ρ}^{μ} b^{v} d x^{ρ}

, or, at the same infinitesimal order,

b^{μ} = a^{μ} - Γ_{ν ρ}^{μ} a^{v} d x^{ρ} .

Levi-Civita thus obtained the intrinsic expression of his newly defined parallel displacement and found it to be identical with the variation he had found in Riemann's Commentatio.

Now equipped with this new geometrical concept, Levi-Civita used it to interpret his expression (88) of the Riemann curvature geometrically (1867,

§ 17 - 18)

. For this purpose, he considered the quasiparallelogram obtained by drawing two geodesics from the points

x

and

x + d x

at a constant angle with

d x

and by marking on these geodesic the points

x^{'}

and

x^{'} + d x^{'}

situated at the same distance

s

from their respective origins (see Figure 7).

To second order in

s

, we have

\begin{aligned} x^{' μ} = x^{μ} + s u^{μ} - \frac{1}{2} s^{2} Γ_{v ρ}^{μ} u^{v} u^{ρ} and \\ x^{' μ} + d x^{' μ} = x^{μ} + d x^{μ} + s u^{μ} (x + d x) - \frac{1}{2} s^{2} Γ_{v ρ}^{μ} (x + d x) u^{v} (x + d x) u^{ρ} (x + d x), \end{aligned}

wherein the

u^{μ}

denote tangent unit vectors. Interpreting

d

as the first-order parallel displacement along

d x

when applied to a vector, introducing the first order differential

δ

for changes occurring on the

x x^{'}

line and interpreting this change as the first-order parallel displacement along this line when needed, Levi-Civita rewrites the preceding equations as

x^{'} = x + δ x + \frac{1}{2} δ^{2} x and d x^{'} = d x + d δ x + \frac{1}{2} d δ^{2} x

To second order in

δ

, the conservation of the length of a vector by parallel displacement along

x x^{'}

gives

g_{μ ν} (x^{'}) d x^{' μ} d x^{' ν} = g_{μ ν} (x) ξ^{μ} ξ^{ν}, with ξ = d x^{'} - δ d x^{'} - \frac{1}{2} δ^{2} d x^{'}

To first order in

d

and second order in

δ

, and using

d δ x = δ d x

, we have

ξ = d x + \frac{1}{2} (d δ - δ d) δ x

and

δ d s^{2} = g_{μ ν} (x^{'}) d x^{' μ} d x^{' ν} - g_{μ ν} (x) d x^{μ} d x^{ν} = g_{μ ν} (x) d x^{μ} (d δ - δ d) δ x^{ν}

Using Eq. (88), this gives

δ d s^{2} = g_{μ τ} d x^{τ} R^{μ}_{ν ρ σ} δ x^{ν} δ x^{ρ} d x^{σ} = - R_{μ ν ρ σ} d x^{μ} δ x^{ν} d x^{ρ} δ x^{σ}

for the difference of length between the two small sides of the quasi-parallelogram.

^{69}

Levi-Civita thus obtained a new geometric interpretation of Riemann's curvature, based on the construction of a quasi-parallelogram by parallel transport. As he tells us in his introduction (1917, 174), he originally believed that Riemann had the same interpretation in mind when he gave the calculation based on the

X

formula of the Commentatio. After closer inspection, Levi-Civita realized that he could not find any trace of the quasi-parallelogram in Riemann's text nor in Dedekind's reconstruction. Worse, he judged that Riemann's

X

should identically vanish because of the identities (89) that express the conservation of length and angles during parallel displacement. He commented (1917, 202):

Probably, in Riemann's

X

there is but some writing defect that conceals the underlying concept. I flatter myself that I have substantially reconstructed this concept, but I could not adjust the symbols. If that can be done, it will be good to give full justice to Riemann's genius in this respect too.

^{70}

As we saw, Riemann's expression does admit a geometrical interpretation (see Figure 2). But the relevant geometric figure differs from Levi-Civita's parallelogram and the associated meaning of the

d

and

δ

symbols is also different. It remains plausible that Riemann arrived at this expression by mostly algebraic considerations.

4. Conclusions

Riemann plausibly obtained his concept of curvature as well as its precise analytical expression by generalizing Gauss's notion of geodetic distortion with the benefit of normal geodesic coordinates, which are themselves a generalization of Gauss's geodesic polar coordinates. The calculations found in the last section of the Commentatio, though not dictated by geometrical considerations, were informed by partial geometrical interpretation of intermediate steps and equations. A certain invariant expression, which I called

X

and which Riemann introduced to show the covariant character of his anticipation of the Riemann tensor, can also be given a geometric interpretation, though not one Riemann is likely to have used in his construction of this expression.

In their commentary to the Commentatio, Dedekind and Weber thoroughly mixed the geometrical and analytical aspects by applying the geometric notion of normal geodesic coordinates to the analytical problem treated in the last section of the Commentatio. Their reconstruction of Riemann's

X

does not match the fragmentary calculations found in the Riemann Nachlass, and it is slightly inconsistent. Their derivation of geodetic distortion in normal geodesic coordinates is nonetheless likely to reproduce features of Riemann's unknown derivation of the same thing at the time of his habilitation lecture. As Dedekind and Weber were not pretending to be doing history, they cannot be blamed for having creatively mixed ideas that Riemann conceived separately.

Christoffel and Lipschitz focused on the algebraic-analytic equivalence problem, and they publicly avoided geometric illustration. The former nonetheless admitted being motivated by the geometric problem of development of a (hyper)surface on another, and both are likely to have relied on analogies with Riemann's or Gauss's more geometric considerations. It would be otherwise difficult to understand how they conceived some of their algebraic maneuvers. That said, their analytical developments mostly depended from the principal context in which they inscribed them: the theory of invariants for Christoffel, and the variational calculus of Lagrangian analytical mechanics for Lipschitz. In these contexts, with or without hidden recourse to geometric analogies, they significantly extended the proto-tensor calculus of the Commentatio, and Lipschitz even solved the mystery of Riemann's

X

in his own manner. Ricci's absolute differential calculus most drastically eliminated extra-analytical considerations, his aim being to provide a universal, absolute differential calculus that would serve many domains of mathematics and physics.

General relativity and Levi-Civita brought physics and geometry back into the interpretation of the Riemann parenthesis or tensor. More than his algebraically inclined predecessors, Levi-Civita felt the artificiality and heaviness of some of the new calculus. He suspected that the formulas found in Riemann's Commentatio depended on hidden geometrical ideas and he hoped to solve the mystery of Riemann's

X

by recreating these ideas. Some of Riemann's formulas involved a peculiar infinitesimal variation of vectors that Levi-Civita knew to preserve angles and length. His interpretation of this variation led him to the concept of parallel transport. Although this concept did not quite solve the

X

mystery, it significantly simplified the apparatus of tensor calculus and differential geometry. It also opened the door to important and vast generalizations of Riemannian geometry.

To sum up, the ambivalent and elliptical character of Riemann's writings on curvature and on the equivalence problem of quadratic differential forms allowed for divergent developments in different contexts including geometry, the theory of invariants, and variational mechanics. Reacting to this diversity, Ricci propounded a decontextualized, universalist tensor analysis, which he called absolute differential calculus. In the end, general relativity and the Riemannian mystery stimulated Levi-Civita's geometrization of Ricci's calculus and thus eased the cross-fertilization of analysis, geometry, and physics.

Acknowledgments

I thank two anonymous reviewers and Tom Archibald for useful suggestions and corrections.

References

Abardia, J., Reventós, A., Rodríguez, C.J., 2012. What did Gauss read in the Appendix? Hist. Math. 39, 292-323.

Beez, R., 1879. Über das Riemann'sche Krümmungsmass höherer Mannigfaltigkeiten. Z. Math. Phys. 24, 1-17,

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- Correspondence to: 83, rue Broca, 75013 Paris, France.
E-mail address: darrigol@paris7.jussieu.fr.
1 "So habe ich noch lebhafte Erinnerung an den ausserordentlichen Eindruck, den Riemanns Gedankengänge damals auf die jungen Mathematiker machten. Vieles erschien uns dunkel und schwerverständlich und doch wieder von unergründlicher Tiefe, wo der heutige Mathematiker, der alle diese Dinge von vornherein in seine Denkweise aufgenommen hat, nur noch die Klarheit und Prägnanz der Auseinandersetzung bewundert."

2 On these circumstances, cf. Dedekind (1876, 517).

3 On the purpose and fate of the Commentatio, cf. Farwell and Knee (1990).
4 This reconstruction of Gauss's route to intrinsic curvature is a guess based on his own hints in Gauss $(1828, 43, 46)$ and on the order of the earlier manuscript (Gauss, 1825), in which the intrinsic character of Gaussian curvature is proved on the basis of its geodetic interpretation (whereas in the final memoir it is proved by analytical means). On Gauss's life, cf. Bühler (1981). On the history of differential geometry in Gauss's times and earlier, cf. Reich (1973). On Gauss's and Riemann's differential geometry, cf. Kline (1972, vol. 3, chap. 37). For a modern assessment of Gauss's contribution, cf. Torretti (1978, 71-82).
5 In the draft $(1825, § 14)$ , Gauss obtained this result by more geometrical means, slicing a finite geodetic triangle into thin sectors (as in a fan) and exploiting spherical geometry on the unit sphere $n^{2} = 1$ .

6 Similar reasoning is found in Gauss $(1825, § 11)$ and in Gauss $(1828, § § 7 - 8)$ .
7 The French mathematician Olinde Rodrigues $(1815 - 1816, 177)$ had already introduced the unit-sphere of unit normals to a surface (in probable analogy with Euler's indicatrix for the principal unit normal to a curve in space) and had proved that the ratio of the corresponding areas were equal to the integral of the product of the two principal curvatures. However, Rodrigues does not seem to have been aware of the invariance of this integral by development (except in the case of developable surfaces, for which it vanishes). Gauss's discovery of this non-evident property plausibly resulted from his starting with the triangular definition of curvature.

8 "Docet itaque analysis in art. praec. explicata, ad inueniendam mensuram curuaturae haud opus esse formulis finitis, quae coordinatas $x, y, z$ , tamquam functiones indeteminatarum $p, q$ exhibeant, sed sufficere expressionem generalem pro magnitudine cuiusuis elementi linearis."
9 The general concept of parallel displacement is not needed here, because along a geodesic of a surface the conditions of constant length and constant angle are sufficient to determine the displacement.
10 This result can be obtained by considering the family of geodesic triangles obtained by drawing two geodesics from the same origin, marking the two points of these geodesics at the respective distances $λ b$ and $λ c$ from the origin, and drawing the geodesic joining these two points. For the length $l$ of this third line, we have $d l = (b \cos ϕ + c \cos ψ) d λ$ (according to the well-known rule for the length variation of a geodesic during a variation of its extremities), if $ϕ$ and $ψ$ denote the lower angles of the $λ$ -triangle. As these angles differ from their planar values $ϕ_{0}$ and $ψ_{0}$ by small quantities, we have $d l = (b \cos ϕ_{0} + c \cos ψ_{0}) d λ - h (ϕ + ψ -$ $ϕ_{0} - ψ_{0}) d λ$ , with $h = b \sin ϕ_{0} = c \cos ψ_{0}$ . Since $ϕ + ψ - ϕ_{0} - ψ_{0} = e (λ) = C λ^{2} S$ , the increment $d l$ exceeds its planar value by $δ d l = - h C S λ^{2} d λ$ . Integrating from $λ = 0$ to $λ = 1$ , we get $δ a = - (1 / 3) h C S$ and $δ (a^{2}) \approx 2 a δ a \approx - (2 / 3) a h C S \approx - (4 / 3) C S^{2}$ .
11 It is possible to give a variant of this reasoning in which the $x^{μ}$ coordinates are arbitrary, by means of third-order expressions of the variation of coordinates along the two sides of the triangle.

12 "Diese Sätze führen dahin, die Theorie der krummen Flächen aus einem neuen Gesichtspuncte zu betrachten, wo sich der Untersuchung ein weites noch ganz unangebauetes Feld öffnet. Wenn man die Flächen nicht als Grenzen von Körpern, sondern als Körper, deren eine Dimension verschwindet, und zugleich als biegsam, aber nicht als dehnbar betrachtet, so begreift man, dass zweyerley wesentlich verschiedene Relationen zu unterschieden sind, theils nämlich solche, die eine bestimmte Form der Fläche im Raume voraussetzen, theils solche, welche von den verschiedenen Formen, die die Fläche annehmen kann, unabhängig
sind. Die letztern sind es, wovon hier die Rede ist: nach dem, was vorhin bemerkt ist, gehört dazu das Krümmungsmass; man sieht aber leicht, dass eben dahin die Betrachtung der aus der Fläche contruierten Figuren, ihrer Winkel, ihres Flächeninhalts und ihrer Totalkrümmung, die Verbindung der Puncte durch kürteste Linien u. dgl. gehört. Alle solche Untersuchungen müssen davon ausgehen, dass die Natur der krummen Fläche an sich durch den Ausdruck eines unbestimmten Linearelements in der Form $\sqrt{E d p^{2} + 2 F d p d q + G d q^{2}}$ gegeben ist."

13 It is not clear whether in 1828 Gauss related this remark with his private anticipation of non-Euclidean geometry: cf. Gray (2006). He might have done so after reading Bolyai in 1832: cf. Abardia et al. (2012).

14 A homogenous differential form of order $k$ is an expression of the kind $\sum_{μ_{1} μ_{2} \dots μ_{k}} a_{μ_{1} μ_{2} \dots μ_{k}} (x) d x_{μ_{1}} d x_{μ_{2}} \dots d x_{μ_{k}}$ , and it corresponds to what we would now call a completely symmetric tensor of order $k$ .

15 "Diese Massverhältnisse lassen sich nur in abstracten Grossenbegriffen untersuchen und im Zusammenhange nur durch Formeln darstellen; unter gewissen Voraussetzungen kann man sie indess in Verhätnisse zerlegen, welche einzeln genommen einer geometrischen Darstellung fähig sind, und hiedurch wird es möglich, die Resultate der Rechnung geometrisch auszudrücken. Es wird daher, um festen Boden zu gewinnen, zwar eine abstracte Untersuchung in Formeln nicht zu vermeiden sein, die Resultate derselben aber werden sich im geometrischen Gewande darstellen lassen. Zu Beidem sind die Grundlagen enthalten in der berühmten Abhandlung des Herrn Geheimen Hofraths Gauss über die krummen Flächen.”

16 On Riemann's lecture and its broader significance in the history of geometry, cf. Scholz (1980, chap. 2), Portnoy (1982), Zund (1983), Scholz (1992), Reich (1994, 26-28), Gray (2007, chap. 18). For modern readings, cf. Spivak (1970-1975, chap. 4, part B), Torretti (1978, 90-101). On Riemann’s biography, cf. Dedekind (1876), Laugwitz (1996).
17 Nowadays, these coordinates are defined through the "exponential map" in the vicinity of the pole.

18 See above, Eqs. (24)-(27). Although Riemann is not likely to have had in hand the full concept of parallel transport, he may have been aware of the length-preserving character of Eq. (14), on which my derivation relies.
19 Riemann's assertion that the value of the sectional curvature in three independent surface directions determines the intrinsic metric properties is incorrect (unless he meant the surface directions to be the principal directions of the quadratic form (29)): cf. Di Scala (2001). It is nevertheless true that the vanishing of the curvature tensor implies the flatness of the manifold. As will be seen below, the first rigorous proofs of this result were given by Lipschitz and Ricci. For a simple proof based on the integrability of parallel displacement, see Eddington (1923, 76-77).

20 This in substance what Dedekind and Weber do in their reconstruction (1876, 389-391) of Riemann's reasoning, as they introduce Gaussian polar coordinates on the geodetic surface.

21 On the Commentatio, its heat propagation context, and its reception, cf. Farwell and Knee (1990). For a modern reading, cf. Spivak (1970-1975, chap, 4, part C).

22 Riemann astutely obtained this equivalence by coordinate-transforming the variational equation $δ \int \sum_{i j} b_{i j} \partial_{i} θ \partial_{j} θ d^{3} s +$ $\int 2 K δ θ (\partial θ / \partial t) d^{3} s = 0$ from which Eq. (32) derives. From a modern point of view, the equivalence immediately results from the invariant expression $Δ θ = g^{- 1 / 2} \partial_{μ} (g^{1 / 2} g^{μ ν} \partial_{ν} θ)$ of the Laplacian on a metric manifold (with $g = \det [g_{μ ν}]$ ). Indeed, for $b^{μ ν} = g^{1 / 2} g^{μ ν}$ and for the matrices $β_{μ ν}$ and $γ_{μ ν}$ of minors associated with $b^{μ ν}$ and $g^{μ ν}$ , we have $β_{μ ν} = g γ_{μ ν}$ (the minors being $2 \times 2$ determinants), $γ_{μ ν} g^{v ρ} = δ_{μ}^{ρ} \det [g^{μ ν}]$ , so that $β_{μ ν} = g_{μ ν}$ (remembering that $g^{μ ν}$ is the inverse matrix of $g_{μ ν}$ ).
23 "Et hanc quaestionem infra videbimus eadem fere methodo tractari posse, qua Gauss in theoria superficierum curvarum usus est."

24 Riemann's notation is $a_{ι, ι^{'}}$ for $g_{μ ν}$ and $p_{ι, l^{'}, ι^{''}}$ for $2 Γ_{μ ν ρ}$ .
25 Against Riemann borrowing this geometric path, it could be argued that in 1869 Christoffel derived a generalization of condition (35) by purely algebraic means. However, it will be seen in a moment (below, Section 3.2) that Christoffel himself plausibly relied on geometric intuition.

26 "Ut indoles harum aequationum melius perspiciatur [ [..]".

27 The $X$ name is mine. Riemann's expression of $X$ has an additional factor 4 because his sums are restricted to $μ < ν$ and $σ < ρ$ .

28 "Jam ex hac formatione hujus expressionis sponte patet, mutatis variabilibus independentibus transmutari eam in expressionem a nova forma ipsius $\sum b_{l, l^{'}} d s_{ι} d s_{ι^{'}}$ eadem lege dependentem."
29 In a draft of the Commentatio (folio 20 of Cod Ms B. Riemann 9), Riemann introduced the geometric interpretation of $g_{μ ν}$ before Eq. (43), which indeed could not be introduced without a geometric motivation.

30 "Disquisitiones haece interpretatione quadam geometrica illustrari possunt, quae quamquam conceptibus inusitatis nitatur, tamen obiter eam addigitavisse juvabit."

31 This remark is found in Klein (1926-1927, 2: 157-158).

32 For a modern interpretation of Riemann's variations, cf. Spivak (1970-1975, chap. 6, add. 2).
33 The last equality in Eq. (47) follows by analogy with the well-known relation $(D_{ν} D_{ρ} - D_{ρ} D_{v}) A^{μ} = R_{σ ν ρ}^{μ} A^{σ}$ between the covariant derivation $D_{μ}$ and the curvature tensor. It can also be derived directly from the definitions of $D$ and $Δ$ , using D $δ x = d δ x + Γ_{δ x, d x}, Δ D δ x = δ d δ x + δ (Γ_{δ x, d x}) + Γ_{δ x, D δ x}, δ (Γ_{δ x, d x}) = δ Γ_{δ x, d x} + Γ_{δ δ x, d x} + Γ_{δ x, δ d x}, Γ_{δ x, D δ x} =$ $Γ_{δ x, d δ x} + Γ_{δ x, Γ_{δ x, d x}}$ , and similar expressions for $D Δ δ x$ . Beware that the $d$ and $δ$ here do not have the same meaning as in my earlier geodetic considerations.

34 I call differential expression of order $k$ any polynomial combination of the differentials $d_{i}^{j} x^{μ}$ such that the sum of all $j$ 's in each term is equal to $k$ (the index $i$ labels the differentials and the index $j$ gives their order). For instance, $(a d^{2} x + b d x d y) δ y$ is a differential expression of third order. Lagrange's theorem is more general, for it includes $f$ functions depending on the higher differentials $d^{2} x, d^{3} x$ , etc.

35 In a moment we will see that Lagrange's considerations inspired Lipschitz's discussion of differential invariants.

36 See note 11 above.
37 Another calculation of $X$ , found on folio 2 , is interesting in its involving the covariant $n_{ι} = \sum_{ι^{'}} a_{ι ι^{'}} d s_{ι^{'}}$ , which correspond to our $d x_{μ} = g_{μ ν} d x^{ν}$ .

38 This identity was first published by Ernesto Padova in 1889 as a result he owed to Ricci, and then independently by Bianchi in 1902: cf. Reich $(1994, 75)$ . Riemann's motivation is unclear.
39 "Analytische Untersuchungen zu Bernhard Riemann's Abhandlungen über die Hypothesen, welche der Geometrie zu Grunde liegen." For the circumstances of these notes, cf. Sinaceur (1990), which also includes the text and a French translation of the notes. 40 See the file Cod Ms Riemann 9 in the Göttingen University archive. Weber later borrowed the final manuscript from the French Academy of Sciences: cf. Weber (1876, IV).
41 Beltrami (1868) independently obtained the generalized Laplacian and its invariance, through a variational procedure: cf. Dell'Aglio (1996, 231-235).
45 "welche besonders deshalb von hohem Interesse ist, weil darin Riemann's Untersuchungen über die allgemeinen Eigenschaften der mehrfach ausgedehnten Mannigfaltigkeiten in den Grundzügen niedergelegt sind und eine merkwürdige Verwendung finden."

46 "während die gegenwärtigen Untersuchungen ursprünglich durch die Ausdehnung des Problems der aufeinander abwickelbaren Flächen auf Gebiete von $n$ Dimensionen veranlasst worden sind."

47 In a footnote to Riemann $(1867, 133)$ , Dedekind had expressed his intention to return to the analytical apparatus of Riemann's lecture in a forthcoming essay.

48 On Christoffel's biography, cf. Butzer (1981).
49 For a lucid interpretation of Christoffel's difficult memoir, cf. Ehlers (1981). On the invariant-theoretical context, cf. Klein (1926-1927, 2: 195-198), Burau (1981), Hunger Parshall (1994).

50 The $Γ$ notation belongs to Einstein (with opposite sign) and Weyl (with the same sign). Christoffel used the symbol ${\begin{matrix} v ρ \\ μ \end{matrix}}$ for $Γ_{v ρ}^{μ}$ and $[\begin{matrix} v ρ \\ μ \end{matrix}]$ for $g_{μ σ} Γ_{ν ρ}^{σ}$ .

51 On this aspect of Christoffel's memoir, cf. Reich $(1994, 59 - 65)$ . The notation $D_{λ} T_{μ ν ρ σ \dots}$ for the new tensor is mine. The interpretation of this operation as a generalized derivation is Ricci's: cf. Dell'Aglio (1996).
52 Cf. Ehlers (1981).

53 One of these mentions Christoffel $(1869, 47)$ concerns the developability of surfaces; another $(1869$ , p. 64 $)$ concerns the invalidity of Christoffel's theorem for a surface that can be shifted [verschiebt] into itself (for instance surfaces of constant curvature).
54 The definition is in note 14 , above.

55 On Lipschitz's contribution, cf. Reich (1994, 29-32) and Tazzioli (1994).

56 In a footnote, Lipschitz $(1869, 74)$ writes: "I have doubts on whether the results of Riemann's memoir can be interpreted in a manner agreeing with [my] theorem."

57 On the Lagrangian, mechanical, and variational context, cf. Tazzioli (1994). Felix Klein (1926-1927, 2: 189) argues that because of its mechanical underpinning Lipschitz's contribution was less appreciated by mathematicians than Chrisfoffel's.

58 In conformity with nineteenth-century terminology, I call differential form of order $k$ on a manifold any $k$ -linear form $\sum_{μ ν \dots} a_{μ ν \dots} (x) d_{1} x^{μ} d_{2} x^{ν} \dots$ of the differentials $d_{1} x, d_{2} x, \dots, d_{k} x$ (in modern terms this is a covariant tensor of order $k$ , which further needs to be completely antisymmetric in order to define a differential form in the modern sense).

59 The relevant concept of differential expression is defined in note 34.
61 Lipschitz was neither the first nor the last to geometrize analytical mechanics. On Liouville's earlier attempt, Hertz's later attempt and the relative independence of Lipschitz's, cf. Lützen (1994, 1999, 2005).

62 Cf. Tazzioli (1994).

63 On Ricci’s approach, cf. Tonolo (1961), Struik (1993), Dell'Aglio (1996, 235-256), Bottazini (1999, 243-253). On his calculus and its influence, cf. Reich (1994, 63-110), Giovanelli (2013, §2.2). The first definition of derivazione covarianti is in Ricci-Curbastro $(1887, 202)$ .
64 "Oggetto di questo scritto è di iniziare sulle forme differenziali quadratiche una serie di ricerche, le quali, condotte su concetti puramente analitici, meglio ci addentrino nella conoscenza della loro natura, e sfuggano anche così alle discussioni, a dir vero, alquanto oziose sulla esistenza e sulla natura degli spazi a più di tre dimensioni. Le interpretazioni dettate da analogie geometriche o meccaniche, che a quei risultati si potranno dare, non saranno che illustrazioni di una tale Teoria."

65 "ridurre alquanto l'apparato formale che serve abitualmente ad introdurli e a stabilirne il comportamento covariante.

66 On Levi-Civita and parallel transport, cf. Struik (1989), Reich (1992), Bottazini (1999, 254-256).
67 "Se non erro, è questo il modo più rapido per arrivarvi."

68 "Il quale, sorto inizialmente con questo solo obbiettivo, venne via via ampliandosi per far debito posto anche all'interpretazione geometrica."
69 Eq. (96) is akin to the theorem of geodesic deviation (Levi-Civita, 1927), which relates the covariant variation of the vector connecting two neighboring geodesics to the curvature tensor.

70 "Probablimente, nella $R$ di Riemann, c'è soltanto un qualche vizio di scrittura, che ne vela il concetto. Mi lusingo di aver sotanzialmente ricostruito tale concetto, ma non potei aggiustare il simbolo. Se la cosa è fattibile, sarà bene rendere, anche su questo particolare, piena giustizia al genio di Riemann."